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  • 1. Lectures on Lévy Processes and Stochastic Calculus (Koc University) Lecture 4: Stochastic Integration and Itô’s Formula David Applebaum School of Mathematics and Statistics, University of Sheffield, UK 8th December 2011Dave Applebaum (Sheffield UK) Lecture 4 December 2011 1 / 58
  • 2. Stochastic IntegrationWe next give a rather rapid account of stochastic integration in a formsuitable for application to Lévy processes.Let X = M + C be a semimartingale. The problem of stochasticintegration is to make sense of objects of the form t t t F (s)dX (s) := F (s)dM(s) + F (s)dC(s). 0 0 0The second integral can be well-defined using the usualLebesgue-Stieltjes approach. The first one cannot - indeed if M is acontinuous martingale of finite variation, then M is a.s. constant. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 2 / 58
  • 3. Stochastic IntegrationWe next give a rather rapid account of stochastic integration in a formsuitable for application to Lévy processes.Let X = M + C be a semimartingale. The problem of stochasticintegration is to make sense of objects of the form t t t F (s)dX (s) := F (s)dM(s) + F (s)dC(s). 0 0 0The second integral can be well-defined using the usualLebesgue-Stieltjes approach. The first one cannot - indeed if M is acontinuous martingale of finite variation, then M is a.s. constant. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 2 / 58
  • 4. Stochastic IntegrationWe next give a rather rapid account of stochastic integration in a formsuitable for application to Lévy processes.Let X = M + C be a semimartingale. The problem of stochasticintegration is to make sense of objects of the form t t t F (s)dX (s) := F (s)dM(s) + F (s)dC(s). 0 0 0The second integral can be well-defined using the usualLebesgue-Stieltjes approach. The first one cannot - indeed if M is acontinuous martingale of finite variation, then M is a.s. constant. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 2 / 58
  • 5. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 6. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 7. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 8. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 9. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 10. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 11. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 12. Refer to the martingale part of the Lévy-Itô decomposition. Define a“martingale-valued measure” by ˜ M(t, E) = B(t)δ0 (E) + N(t, E − {0}),for E ∈ B(Rd ), where B = (B(t), t ≥ 0) is a one-dimensional Brownianmotion. The following key properties then hold:- M((s, t], E) = M(t, E) − M(s, E) is independent of Fs , for 0 ≤ s < t < ∞. E(M((s, t], E)) = 0. E(M((s, t], E)2 ) = ρ((s, t], E) where ρ((s, t], E) = (t − s)(δ0 (E) + ν(E − {0})). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 3 / 58
  • 13. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 14. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 15. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 16. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 17. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 18. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 19. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 20. We’re going to unify the usual stochastic integral with the Poissonintegral, by defining: t t t F (s, x)M(ds, dx) := G(s)dB(s) + ˜ F (s, x)N(ds, dx). 0 E 0 0 E−{0}where G(s) = F (s, 0). Of course, we need some conditions on theclass of integrands:-Fix E ∈ B(Rd ) and 0 < T < ∞ and let P denote the smallest σ-algebrawith respect to which all mappings F : [0, T ] × E × Ω → R satisfying (1)and (2) below are measurable. 1 For each 0 ≤ t ≤ T , the mapping (x, ω) → F (t, x, ω) is B(E) ⊗ Ft measurable, 2 For each x ∈ E, ω ∈ Ω, the mapping t → F (t, x, ω) is left continuous. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 4 / 58
  • 21. We call P the predictable σ-algebra. A P-measurable mappingG : [0, T ] × E × Ω → R is then said to be predictable. The definitionclearly extends naturally to the case where [0, T ] is replaced by R+ .Note that by (1), if G is predictable then the process t → G(t, x, ·) isadapted, for each x ∈ E. If G satisfies (1) and is left continuous then itis clearly predictable. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 5 / 58
  • 22. We call P the predictable σ-algebra. A P-measurable mappingG : [0, T ] × E × Ω → R is then said to be predictable. The definitionclearly extends naturally to the case where [0, T ] is replaced by R+ .Note that by (1), if G is predictable then the process t → G(t, x, ·) isadapted, for each x ∈ E. If G satisfies (1) and is left continuous then itis clearly predictable. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 5 / 58
  • 23. We call P the predictable σ-algebra. A P-measurable mappingG : [0, T ] × E × Ω → R is then said to be predictable. The definitionclearly extends naturally to the case where [0, T ] is replaced by R+ .Note that by (1), if G is predictable then the process t → G(t, x, ·) isadapted, for each x ∈ E. If G satisfies (1) and is left continuous then itis clearly predictable. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 5 / 58
  • 24. We call P the predictable σ-algebra. A P-measurable mappingG : [0, T ] × E × Ω → R is then said to be predictable. The definitionclearly extends naturally to the case where [0, T ] is replaced by R+ .Note that by (1), if G is predictable then the process t → G(t, x, ·) isadapted, for each x ∈ E. If G satisfies (1) and is left continuous then itis clearly predictable. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 5 / 58
  • 25. We call P the predictable σ-algebra. A P-measurable mappingG : [0, T ] × E × Ω → R is then said to be predictable. The definitionclearly extends naturally to the case where [0, T ] is replaced by R+ .Note that by (1), if G is predictable then the process t → G(t, x, ·) isadapted, for each x ∈ E. If G satisfies (1) and is left continuous then itis clearly predictable. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 5 / 58
  • 26. Define H2 (T , E) to be the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P and which satisfy the following conditions: F is predictable, T E(|F (t, x)|2 )ρ(dt, dx) < ∞. 0 EIt can be shown that H2 (T , E) is a real Hilbert space with respect to Tthe inner product < F , G >T ,ρ = 0 E E((F (t, x), G(t, x)))ρ(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 6 / 58
  • 27. Define H2 (T , E) to be the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P and which satisfy the following conditions: F is predictable, T E(|F (t, x)|2 )ρ(dt, dx) < ∞. 0 EIt can be shown that H2 (T , E) is a real Hilbert space with respect to Tthe inner product < F , G >T ,ρ = 0 E E((F (t, x), G(t, x)))ρ(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 6 / 58
  • 28. Define H2 (T , E) to be the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P and which satisfy the following conditions: F is predictable, T E(|F (t, x)|2 )ρ(dt, dx) < ∞. 0 EIt can be shown that H2 (T , E) is a real Hilbert space with respect to Tthe inner product < F , G >T ,ρ = 0 E E((F (t, x), G(t, x)))ρ(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 6 / 58
  • 29. Define H2 (T , E) to be the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P and which satisfy the following conditions: F is predictable, T E(|F (t, x)|2 )ρ(dt, dx) < ∞. 0 EIt can be shown that H2 (T , E) is a real Hilbert space with respect to Tthe inner product < F , G >T ,ρ = 0 E E((F (t, x), G(t, x)))ρ(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 6 / 58
  • 30. Define S(T , E) to be the linear space of all simple processes inH2 (T , E),where F is simple if for some m, n ∈ N, there exists0 ≤ t1 ≤ t2 ≤ · · · ≤ tm+1 = T and there exists a family of disjoint Borelsubsets A1 , A2 , . . . , An of E with each ν(Ai ) < ∞ such that m n F = Fj,k 1(tj ,tj+1 ] 1Ak , j=1 k =1where each Fj,k is a bounded Ftj -measurable random variable. Notethat F is left continuous and B(E) ⊗ Ft measurable, hence it ispredictable. An important fact is that S(T , E) is dense in H2 (T , E), Dave Applebaum (Sheffield UK) Lecture 4 December 2011 7 / 58
  • 31. Define S(T , E) to be the linear space of all simple processes inH2 (T , E),where F is simple if for some m, n ∈ N, there exists0 ≤ t1 ≤ t2 ≤ · · · ≤ tm+1 = T and there exists a family of disjoint Borelsubsets A1 , A2 , . . . , An of E with each ν(Ai ) < ∞ such that m n F = Fj,k 1(tj ,tj+1 ] 1Ak , j=1 k =1where each Fj,k is a bounded Ftj -measurable random variable. Notethat F is left continuous and B(E) ⊗ Ft measurable, hence it ispredictable. An important fact is that S(T , E) is dense in H2 (T , E), Dave Applebaum (Sheffield UK) Lecture 4 December 2011 7 / 58
  • 32. Define S(T , E) to be the linear space of all simple processes inH2 (T , E),where F is simple if for some m, n ∈ N, there exists0 ≤ t1 ≤ t2 ≤ · · · ≤ tm+1 = T and there exists a family of disjoint Borelsubsets A1 , A2 , . . . , An of E with each ν(Ai ) < ∞ such that m n F = Fj,k 1(tj ,tj+1 ] 1Ak , j=1 k =1where each Fj,k is a bounded Ftj -measurable random variable. Notethat F is left continuous and B(E) ⊗ Ft measurable, hence it ispredictable. An important fact is that S(T , E) is dense in H2 (T , E), Dave Applebaum (Sheffield UK) Lecture 4 December 2011 7 / 58
  • 33. Define S(T , E) to be the linear space of all simple processes inH2 (T , E),where F is simple if for some m, n ∈ N, there exists0 ≤ t1 ≤ t2 ≤ · · · ≤ tm+1 = T and there exists a family of disjoint Borelsubsets A1 , A2 , . . . , An of E with each ν(Ai ) < ∞ such that m n F = Fj,k 1(tj ,tj+1 ] 1Ak , j=1 k =1where each Fj,k is a bounded Ftj -measurable random variable. Notethat F is left continuous and B(E) ⊗ Ft measurable, hence it ispredictable. An important fact is that S(T , E) is dense in H2 (T , E), Dave Applebaum (Sheffield UK) Lecture 4 December 2011 7 / 58
  • 34. Define S(T , E) to be the linear space of all simple processes inH2 (T , E),where F is simple if for some m, n ∈ N, there exists0 ≤ t1 ≤ t2 ≤ · · · ≤ tm+1 = T and there exists a family of disjoint Borelsubsets A1 , A2 , . . . , An of E with each ν(Ai ) < ∞ such that m n F = Fj,k 1(tj ,tj+1 ] 1Ak , j=1 k =1where each Fj,k is a bounded Ftj -measurable random variable. Notethat F is left continuous and B(E) ⊗ Ft measurable, hence it ispredictable. An important fact is that S(T , E) is dense in H2 (T , E), Dave Applebaum (Sheffield UK) Lecture 4 December 2011 7 / 58
  • 35. One of Itô’s greatest achievements was the definition of the stochasticintegral IT (F ), for F simple, by separating the “past” from the “future”within the Riemann sum:- m n IT (F ) = Fj,k M((tj , tj+1 ], Ak ), (0.1) j=1 k =1so on each time interval [tj , tj+1 ], Fj,k encapsulates informationobtained by time tj , while M gives the innovation into the future (tj , tj+1 ]. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 8 / 58
  • 36. One of Itô’s greatest achievements was the definition of the stochasticintegral IT (F ), for F simple, by separating the “past” from the “future”within the Riemann sum:- m n IT (F ) = Fj,k M((tj , tj+1 ], Ak ), (0.1) j=1 k =1so on each time interval [tj , tj+1 ], Fj,k encapsulates informationobtained by time tj , while M gives the innovation into the future (tj , tj+1 ]. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 8 / 58
  • 37. One of Itô’s greatest achievements was the definition of the stochasticintegral IT (F ), for F simple, by separating the “past” from the “future”within the Riemann sum:- m n IT (F ) = Fj,k M((tj , tj+1 ], Ak ), (0.1) j=1 k =1so on each time interval [tj , tj+1 ], Fj,k encapsulates informationobtained by time tj , while M gives the innovation into the future (tj , tj+1 ]. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 8 / 58
  • 38. LemmaFor each T ≥ 0, F ∈ S(T , E), T E(IT (F )) = 0, E(IT (F )2 ) = E(|F (t, x)|2 )ρ(dt, dx). 0 EProof. E(IT (F )) = 0 is a straightforward application of linearity andindependence. The second result is quite messy - we lose nothingimportant by just looking at the Brownian case, with d = 1. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 9 / 58
  • 39. LemmaFor each T ≥ 0, F ∈ S(T , E), T E(IT (F )) = 0, E(IT (F )2 ) = E(|F (t, x)|2 )ρ(dt, dx). 0 EProof. E(IT (F )) = 0 is a straightforward application of linearity andindependence. The second result is quite messy - we lose nothingimportant by just looking at the Brownian case, with d = 1. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 9 / 58
  • 40. LemmaFor each T ≥ 0, F ∈ S(T , E), T E(IT (F )) = 0, E(IT (F )2 ) = E(|F (t, x)|2 )ρ(dt, dx). 0 EProof. E(IT (F )) = 0 is a straightforward application of linearity andindependence. The second result is quite messy - we lose nothingimportant by just looking at the Brownian case, with d = 1. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 9 / 58
  • 41. m mSo let F (t) := j=1 Fj 1(tj ,tj+1 ] , then IT (F ) = j=1 Fj (B(tj+1 ) − B(tj )), m m IT (F )2 = Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp )). j=1 p=1Now fix j and split the second sum into three pieces - corresponding top < j, p = j and p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 10 / 58
  • 42. m mSo let F (t) := j=1 Fj 1(tj ,tj+1 ] , then IT (F ) = j=1 Fj (B(tj+1 ) − B(tj )), m m IT (F )2 = Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp )). j=1 p=1Now fix j and split the second sum into three pieces - corresponding top < j, p = j and p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 10 / 58
  • 43. m mSo let F (t) := j=1 Fj 1(tj ,tj+1 ] , then IT (F ) = j=1 Fj (B(tj+1 ) − B(tj )), m m IT (F )2 = Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp )). j=1 p=1Now fix j and split the second sum into three pieces - corresponding top < j, p = j and p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 10 / 58
  • 44. m mSo let F (t) := j=1 Fj 1(tj ,tj+1 ] , then IT (F ) = j=1 Fj (B(tj+1 ) − B(tj )), m m IT (F )2 = Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp )). j=1 p=1Now fix j and split the second sum into three pieces - corresponding top < j, p = j and p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 10 / 58
  • 45. When p < j, Fj Fp (B(tp+1 ) − B(tp )) ∈ Ftj which is independent ofB(tj+1 ) − B(tj ), E[Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp ))] = E[Fj Fp (B(tp+1 ) − B(tp ))]E(B(tj+1 ) − B(tj )) = 0.Exactly the same argument works when p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 11 / 58
  • 46. When p < j, Fj Fp (B(tp+1 ) − B(tp )) ∈ Ftj which is independent ofB(tj+1 ) − B(tj ), E[Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp ))] = E[Fj Fp (B(tp+1 ) − B(tp ))]E(B(tj+1 ) − B(tj )) = 0.Exactly the same argument works when p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 11 / 58
  • 47. When p < j, Fj Fp (B(tp+1 ) − B(tp )) ∈ Ftj which is independent ofB(tj+1 ) − B(tj ), E[Fj Fp (B(tj+1 ) − B(tj ))(B(tp+1 ) − B(tp ))] = E[Fj Fp (B(tp+1 ) − B(tp ))]E(B(tj+1 ) − B(tj )) = 0.Exactly the same argument works when p > j. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 11 / 58
  • 48. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 49. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 50. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 51. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 52. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 53. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 54. What remains is the case p = j, and by independence again m E(IT (F )2 ) = E(Fj2 )E(B(tj+1 ) − B(tj ))2 j=1 m = E(Fj2 )(tj+1 − tj ). 2 j=1We deduce from Lemma 1 that IT is a linear isometry from S(T , E) intoL2 (Ω, F, P),and hence it extends to an isometric embedding of thewhole of H2 (T , E) into L2 (Ω, F, P). We continue to denote thisextension as IT and we call IT (F ) the (Itô) stochastic integral ofF ∈ H2 (T , E). When convenient, we will use the Leibniz notation TIT (F ) := 0 E F (t, x)M(dt, dx). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 12 / 58
  • 55. TSo for predictable F satisfying 0 E E(|F (s, x)|2 )ρ(ds, dx) < ∞, wecan find a sequence (Fn , n ∈ N) of simple processes such that T T F (t, x)M(dt, dx) = lim Fn (t, x)M(dt, dx). 0 E n→∞ 0 EThe limit is taken in the L2 -sense and is independent of the choice ofapproximating sequence. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 13 / 58
  • 56. TSo for predictable F satisfying 0 E E(|F (s, x)|2 )ρ(ds, dx) < ∞, wecan find a sequence (Fn , n ∈ N) of simple processes such that T T F (t, x)M(dt, dx) = lim Fn (t, x)M(dt, dx). 0 E n→∞ 0 EThe limit is taken in the L2 -sense and is independent of the choice ofapproximating sequence. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 13 / 58
  • 57. TSo for predictable F satisfying 0 E E(|F (s, x)|2 )ρ(ds, dx) < ∞, wecan find a sequence (Fn , n ∈ N) of simple processes such that T T F (t, x)M(dt, dx) = lim Fn (t, x)M(dt, dx). 0 E n→∞ 0 EThe limit is taken in the L2 -sense and is independent of the choice ofapproximating sequence. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 13 / 58
  • 58. TSo for predictable F satisfying 0 E E(|F (s, x)|2 )ρ(ds, dx) < ∞, wecan find a sequence (Fn , n ∈ N) of simple processes such that T T F (t, x)M(dt, dx) = lim Fn (t, x)M(dt, dx). 0 E n→∞ 0 EThe limit is taken in the L2 -sense and is independent of the choice ofapproximating sequence. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 13 / 58
  • 59. The following theorem summarises some useful properties of thestochastic integral.TheoremIf F , G ∈ H2 (T , E) and α, β ∈ R, then : 1 IT (αF + βG) = αIT (F ) + βIT (G). T 2 E(IT (F )) = 0, E(IT (F )2 ) = 0 E E(|F (t, x)|2 )ρ(dt, dx). 3 (It (F ), t ≥ 0) is Ft -adapted. 4 (It (F ), t ≥ 0) is a square-integrable martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 14 / 58
  • 60. The following theorem summarises some useful properties of thestochastic integral.TheoremIf F , G ∈ H2 (T , E) and α, β ∈ R, then : 1 IT (αF + βG) = αIT (F ) + βIT (G). T 2 E(IT (F )) = 0, E(IT (F )2 ) = 0 E E(|F (t, x)|2 )ρ(dt, dx). 3 (It (F ), t ≥ 0) is Ft -adapted. 4 (It (F ), t ≥ 0) is a square-integrable martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 14 / 58
  • 61. The following theorem summarises some useful properties of thestochastic integral.TheoremIf F , G ∈ H2 (T , E) and α, β ∈ R, then : 1 IT (αF + βG) = αIT (F ) + βIT (G). T 2 E(IT (F )) = 0, E(IT (F )2 ) = 0 E E(|F (t, x)|2 )ρ(dt, dx). 3 (It (F ), t ≥ 0) is Ft -adapted. 4 (It (F ), t ≥ 0) is a square-integrable martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 14 / 58
  • 62. The following theorem summarises some useful properties of thestochastic integral.TheoremIf F , G ∈ H2 (T , E) and α, β ∈ R, then : 1 IT (αF + βG) = αIT (F ) + βIT (G). T 2 E(IT (F )) = 0, E(IT (F )2 ) = 0 E E(|F (t, x)|2 )ρ(dt, dx). 3 (It (F ), t ≥ 0) is Ft -adapted. 4 (It (F ), t ≥ 0) is a square-integrable martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 14 / 58
  • 63. The following theorem summarises some useful properties of thestochastic integral.TheoremIf F , G ∈ H2 (T , E) and α, β ∈ R, then : 1 IT (αF + βG) = αIT (F ) + βIT (G). T 2 E(IT (F )) = 0, E(IT (F )2 ) = 0 E E(|F (t, x)|2 )ρ(dt, dx). 3 (It (F ), t ≥ 0) is Ft -adapted. 4 (It (F ), t ≥ 0) is a square-integrable martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 14 / 58
  • 64. The following theorem summarises some useful properties of thestochastic integral.TheoremIf F , G ∈ H2 (T , E) and α, β ∈ R, then : 1 IT (αF + βG) = αIT (F ) + βIT (G). T 2 E(IT (F )) = 0, E(IT (F )2 ) = 0 E E(|F (t, x)|2 )ρ(dt, dx). 3 (It (F ), t ≥ 0) is Ft -adapted. 4 (It (F ), t ≥ 0) is a square-integrable martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 14 / 58
  • 65. Proof. (1) and (2) are easy.For (3), let (Fn , n ∈ N) be a sequence in S(T , E) converging to F ; theneach process (It (Fn ), t ≥ 0) is clearly adapted. Since eachIt (Fn ) → It (F ) in L2 as n → ∞, we can find a subsequence(Fnk , nk ∈ N) such that It (Fnk ) → It (F ) a.s. as nk → ∞, and therequired result follows. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 15 / 58
  • 66. Proof. (1) and (2) are easy.For (3), let (Fn , n ∈ N) be a sequence in S(T , E) converging to F ; theneach process (It (Fn ), t ≥ 0) is clearly adapted. Since eachIt (Fn ) → It (F ) in L2 as n → ∞, we can find a subsequence(Fnk , nk ∈ N) such that It (Fnk ) → It (F ) a.s. as nk → ∞, and therequired result follows. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 15 / 58
  • 67. Proof. (1) and (2) are easy.For (3), let (Fn , n ∈ N) be a sequence in S(T , E) converging to F ; theneach process (It (Fn ), t ≥ 0) is clearly adapted. Since eachIt (Fn ) → It (F ) in L2 as n → ∞, we can find a subsequence(Fnk , nk ∈ N) such that It (Fnk ) → It (F ) a.s. as nk → ∞, and therequired result follows. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 15 / 58
  • 68. Proof. (1) and (2) are easy.For (3), let (Fn , n ∈ N) be a sequence in S(T , E) converging to F ; theneach process (It (Fn ), t ≥ 0) is clearly adapted. Since eachIt (Fn ) → It (F ) in L2 as n → ∞, we can find a subsequence(Fnk , nk ∈ N) such that It (Fnk ) → It (F ) a.s. as nk → ∞, and therequired result follows. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 15 / 58
  • 69. Proof. (1) and (2) are easy.For (3), let (Fn , n ∈ N) be a sequence in S(T , E) converging to F ; theneach process (It (Fn ), t ≥ 0) is clearly adapted. Since eachIt (Fn ) → It (F ) in L2 as n → ∞, we can find a subsequence(Fnk , nk ∈ N) such that It (Fnk ) → It (F ) a.s. as nk → ∞, and therequired result follows. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 15 / 58
  • 70. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 71. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 72. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 73. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 74. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 75. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 76. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 77. (4) Let F ∈ S(T , E) and (without loss of generality) choose0 < s = tl < tl+1 < t. Then it is easy to see that It (F ) = Is (F ) + Is,t (F )and hence Es (It (F )) = Is (F ) + Es (Is,t (F )) by (3). However,   m n Es (Is,t (F )) = Es  Fj,k M((tj , tj+1 ], Ak ) j=l+1 k =1 n n = Es (Fj,k )E(M((tj , tj+1 ], Ak )) = 0. j=l+1 k =1The result now follows by the continuity of Es in L2 . Indeed, let(Fn , n ∈ N) be a sequence in S(T , E) converging to F ; then we have||Es (It (F )) − Es (It (Fn ))||2 ≤ ||It (F ) − It (Fn )||2 = ||F − Fn ||T ,ρ → 0 as n → ∞. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 16 / 58
  • 78. We can extend the stochastic integral IT (F ) to integrands inP2 (T , E).This is the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P, and which satisfy the following conditions: F is predictable. T P 0 E |F (t, x)|2 ρ(dt, dx) < ∞ = 1.If F ∈ P2 (T , E), (It (F ), t ≥ 0) is always a local martingale, but notnecessarily a martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 17 / 58
  • 79. We can extend the stochastic integral IT (F ) to integrands inP2 (T , E).This is the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P, and which satisfy the following conditions: F is predictable. T P 0 E |F (t, x)|2 ρ(dt, dx) < ∞ = 1.If F ∈ P2 (T , E), (It (F ), t ≥ 0) is always a local martingale, but notnecessarily a martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 17 / 58
  • 80. We can extend the stochastic integral IT (F ) to integrands inP2 (T , E).This is the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P, and which satisfy the following conditions: F is predictable. T P 0 E |F (t, x)|2 ρ(dt, dx) < ∞ = 1.If F ∈ P2 (T , E), (It (F ), t ≥ 0) is always a local martingale, but notnecessarily a martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 17 / 58
  • 81. We can extend the stochastic integral IT (F ) to integrands inP2 (T , E).This is the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P, and which satisfy the following conditions: F is predictable. T P 0 E |F (t, x)|2 ρ(dt, dx) < ∞ = 1.If F ∈ P2 (T , E), (It (F ), t ≥ 0) is always a local martingale, but notnecessarily a martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 17 / 58
  • 82. We can extend the stochastic integral IT (F ) to integrands inP2 (T , E).This is the linear space of all equivalence classes ofmappings F : [0, T ] × E × Ω → R which coincide almost everywherewith respect to ρ × P, and which satisfy the following conditions: F is predictable. T P 0 E |F (t, x)|2 ρ(dt, dx) < ∞ = 1.If F ∈ P2 (T , E), (It (F ), t ≥ 0) is always a local martingale, but notnecessarily a martingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 17 / 58
  • 83. Poisson Stochastic IntegralsLet A be an arbitrary Borel set in Rd − {0} which is bounded below,and introduce the compound Poisson process P = (P(t), t ≥ 0), whereeach P(t) = A xN(t, dx). Let K be a predictable mapping, thengeneralising the earlier Poisson integrals, we define T K (t, x)N(dt, dx) = K (u, ∆P(u))1A (∆P(u)), (0.2) 0 A 0≤u≤Tas a random finite sum. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 18 / 58
  • 84. Poisson Stochastic IntegralsLet A be an arbitrary Borel set in Rd − {0} which is bounded below,and introduce the compound Poisson process P = (P(t), t ≥ 0), whereeach P(t) = A xN(t, dx). Let K be a predictable mapping, thengeneralising the earlier Poisson integrals, we define T K (t, x)N(dt, dx) = K (u, ∆P(u))1A (∆P(u)), (0.2) 0 A 0≤u≤Tas a random finite sum. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 18 / 58
  • 85. Poisson Stochastic IntegralsLet A be an arbitrary Borel set in Rd − {0} which is bounded below,and introduce the compound Poisson process P = (P(t), t ≥ 0), whereeach P(t) = A xN(t, dx). Let K be a predictable mapping, thengeneralising the earlier Poisson integrals, we define T K (t, x)N(dt, dx) = K (u, ∆P(u))1A (∆P(u)), (0.2) 0 A 0≤u≤Tas a random finite sum. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 18 / 58
  • 86. Poisson Stochastic IntegralsLet A be an arbitrary Borel set in Rd − {0} which is bounded below,and introduce the compound Poisson process P = (P(t), t ≥ 0), whereeach P(t) = A xN(t, dx). Let K be a predictable mapping, thengeneralising the earlier Poisson integrals, we define T K (t, x)N(dt, dx) = K (u, ∆P(u))1A (∆P(u)), (0.2) 0 A 0≤u≤Tas a random finite sum. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 18 / 58
  • 87. In particular, if H satisfies the square-integrability condition givenabove, we may then define, for each 1 ≤ i ≤ d, T ˜ H i (t, x)N(dt, dx) 0 A T T := H i (t, x)N(dt, dx) − H i (t, x)ν(dx)dt. 0 A 0 AThe definition (0.2) can, in principle, be used to define stochasticintegrals for a more general class of integrands than we have beenconsidering. For simplicity, let N = (N(t), t ≥ 0) be a Poisson processof intensity 1 and let f : R → R, then we may define t f (N(s))dN(s) = f (N(s−) + ∆N(s))∆N(s). 0 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 19 / 58
  • 88. In particular, if H satisfies the square-integrability condition givenabove, we may then define, for each 1 ≤ i ≤ d, T ˜ H i (t, x)N(dt, dx) 0 A T T := H i (t, x)N(dt, dx) − H i (t, x)ν(dx)dt. 0 A 0 AThe definition (0.2) can, in principle, be used to define stochasticintegrals for a more general class of integrands than we have beenconsidering. For simplicity, let N = (N(t), t ≥ 0) be a Poisson processof intensity 1 and let f : R → R, then we may define t f (N(s))dN(s) = f (N(s−) + ∆N(s))∆N(s). 0 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 19 / 58
  • 89. In particular, if H satisfies the square-integrability condition givenabove, we may then define, for each 1 ≤ i ≤ d, T ˜ H i (t, x)N(dt, dx) 0 A T T := H i (t, x)N(dt, dx) − H i (t, x)ν(dx)dt. 0 A 0 AThe definition (0.2) can, in principle, be used to define stochasticintegrals for a more general class of integrands than we have beenconsidering. For simplicity, let N = (N(t), t ≥ 0) be a Poisson processof intensity 1 and let f : R → R, then we may define t f (N(s))dN(s) = f (N(s−) + ∆N(s))∆N(s). 0 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 19 / 58
  • 90. Lévy-type stochastic integrals ˆWe take E = B − {0} = {x ∈ Rd ; 0 < |x| < 1} throughout thissubsection. We say that an Rd -valued stochastic processY = (Y (t), t ≥ 0) is a Lévy-type stochastic integral if it can be writtenin the following form for each 1 ≤ i ≤ d, t ≥ 0, t t Y i (t) = Y i (0) + Gi (s)ds + Fji (s)dB j (s) 0 0 t t + ˜ H i (s, x)N(ds, dx) + K i (s, x)N(ds, dx), 0 |x|<1 0 |x|≥1where for each 11 ≤ i ≤ d, 1 ≤ j ≤ m, t ≥ 0, |Gi | 2 , Fji ∈ P2 (T ), H i ∈ P2 (T , E) and K ispredictable. B is an m-dimensional standard Brownian motion and N isan independent Poisson random measure on R+ × (Rd − {0}) with ˜compensator N and intensity measure ν, which we will assume is aLévy measure. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 20 / 58
  • 91. Lévy-type stochastic integrals ˆWe take E = B − {0} = {x ∈ Rd ; 0 < |x| < 1} throughout thissubsection. We say that an Rd -valued stochastic processY = (Y (t), t ≥ 0) is a Lévy-type stochastic integral if it can be writtenin the following form for each 1 ≤ i ≤ d, t ≥ 0, t t Y i (t) = Y i (0) + Gi (s)ds + Fji (s)dB j (s) 0 0 t t + ˜ H i (s, x)N(ds, dx) + K i (s, x)N(ds, dx), 0 |x|<1 0 |x|≥1where for each 11 ≤ i ≤ d, 1 ≤ j ≤ m, t ≥ 0, |Gi | 2 , Fji ∈ P2 (T ), H i ∈ P2 (T , E) and K ispredictable. B is an m-dimensional standard Brownian motion and N isan independent Poisson random measure on R+ × (Rd − {0}) with ˜compensator N and intensity measure ν, which we will assume is aLévy measure. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 20 / 58
  • 92. Lévy-type stochastic integrals ˆWe take E = B − {0} = {x ∈ Rd ; 0 < |x| < 1} throughout thissubsection. We say that an Rd -valued stochastic processY = (Y (t), t ≥ 0) is a Lévy-type stochastic integral if it can be writtenin the following form for each 1 ≤ i ≤ d, t ≥ 0, t t Y i (t) = Y i (0) + Gi (s)ds + Fji (s)dB j (s) 0 0 t t + ˜ H i (s, x)N(ds, dx) + K i (s, x)N(ds, dx), 0 |x|<1 0 |x|≥1where for each 11 ≤ i ≤ d, 1 ≤ j ≤ m, t ≥ 0, |Gi | 2 , Fji ∈ P2 (T ), H i ∈ P2 (T , E) and K ispredictable. B is an m-dimensional standard Brownian motion and N isan independent Poisson random measure on R+ × (Rd − {0}) with ˜compensator N and intensity measure ν, which we will assume is aLévy measure. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 20 / 58
  • 93. Lévy-type stochastic integrals ˆWe take E = B − {0} = {x ∈ Rd ; 0 < |x| < 1} throughout thissubsection. We say that an Rd -valued stochastic processY = (Y (t), t ≥ 0) is a Lévy-type stochastic integral if it can be writtenin the following form for each 1 ≤ i ≤ d, t ≥ 0, t t Y i (t) = Y i (0) + Gi (s)ds + Fji (s)dB j (s) 0 0 t t + ˜ H i (s, x)N(ds, dx) + K i (s, x)N(ds, dx), 0 |x|<1 0 |x|≥1where for each 11 ≤ i ≤ d, 1 ≤ j ≤ m, t ≥ 0, |Gi | 2 , Fji ∈ P2 (T ), H i ∈ P2 (T , E) and K ispredictable. B is an m-dimensional standard Brownian motion and N isan independent Poisson random measure on R+ × (Rd − {0}) with ˜compensator N and intensity measure ν, which we will assume is aLévy measure. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 20 / 58
  • 94. Lévy-type stochastic integrals ˆWe take E = B − {0} = {x ∈ Rd ; 0 < |x| < 1} throughout thissubsection. We say that an Rd -valued stochastic processY = (Y (t), t ≥ 0) is a Lévy-type stochastic integral if it can be writtenin the following form for each 1 ≤ i ≤ d, t ≥ 0, t t Y i (t) = Y i (0) + Gi (s)ds + Fji (s)dB j (s) 0 0 t t + ˜ H i (s, x)N(ds, dx) + K i (s, x)N(ds, dx), 0 |x|<1 0 |x|≥1where for each 11 ≤ i ≤ d, 1 ≤ j ≤ m, t ≥ 0, |Gi | 2 , Fji ∈ P2 (T ), H i ∈ P2 (T , E) and K ispredictable. B is an m-dimensional standard Brownian motion and N isan independent Poisson random measure on R+ × (Rd − {0}) with ˜compensator N and intensity measure ν, which we will assume is aLévy measure. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 20 / 58
  • 95. We will assume that the random variable Y (0) is F0 -measurable, andthen it is clear that Y is an adapted process.We can often simplify complicated expressions by employing thenotation of stochastic differentials to represent Lévy-type stochasticintegrals. We then write the last expression as ˜ dY (t) = G(t)dt + F (t)dB(t) + H(t, x)N(dt, dx) + K (t, x)N(dt, dx).When we want to particularly emphasise the domains of integrationwith respect to x, we will use an equivalent notation dY (t) = G(t)dt + F (t)dB(t) + ˜ H(t, x)N(dt, dx) |x|<1 + K (t, x)N(dt, dx). |x|≥1Clearly Y is a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 21 / 58
  • 96. We will assume that the random variable Y (0) is F0 -measurable, andthen it is clear that Y is an adapted process.We can often simplify complicated expressions by employing thenotation of stochastic differentials to represent Lévy-type stochasticintegrals. We then write the last expression as ˜ dY (t) = G(t)dt + F (t)dB(t) + H(t, x)N(dt, dx) + K (t, x)N(dt, dx).When we want to particularly emphasise the domains of integrationwith respect to x, we will use an equivalent notation dY (t) = G(t)dt + F (t)dB(t) + ˜ H(t, x)N(dt, dx) |x|<1 + K (t, x)N(dt, dx). |x|≥1Clearly Y is a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 21 / 58
  • 97. We will assume that the random variable Y (0) is F0 -measurable, andthen it is clear that Y is an adapted process.We can often simplify complicated expressions by employing thenotation of stochastic differentials to represent Lévy-type stochasticintegrals. We then write the last expression as ˜ dY (t) = G(t)dt + F (t)dB(t) + H(t, x)N(dt, dx) + K (t, x)N(dt, dx).When we want to particularly emphasise the domains of integrationwith respect to x, we will use an equivalent notation dY (t) = G(t)dt + F (t)dB(t) + ˜ H(t, x)N(dt, dx) |x|<1 + K (t, x)N(dt, dx). |x|≥1Clearly Y is a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 21 / 58
  • 98. We will assume that the random variable Y (0) is F0 -measurable, andthen it is clear that Y is an adapted process.We can often simplify complicated expressions by employing thenotation of stochastic differentials to represent Lévy-type stochasticintegrals. We then write the last expression as ˜ dY (t) = G(t)dt + F (t)dB(t) + H(t, x)N(dt, dx) + K (t, x)N(dt, dx).When we want to particularly emphasise the domains of integrationwith respect to x, we will use an equivalent notation dY (t) = G(t)dt + F (t)dB(t) + ˜ H(t, x)N(dt, dx) |x|<1 + K (t, x)N(dt, dx). |x|≥1Clearly Y is a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 21 / 58
  • 99. We will assume that the random variable Y (0) is F0 -measurable, andthen it is clear that Y is an adapted process.We can often simplify complicated expressions by employing thenotation of stochastic differentials to represent Lévy-type stochasticintegrals. We then write the last expression as ˜ dY (t) = G(t)dt + F (t)dB(t) + H(t, x)N(dt, dx) + K (t, x)N(dt, dx).When we want to particularly emphasise the domains of integrationwith respect to x, we will use an equivalent notation dY (t) = G(t)dt + F (t)dB(t) + ˜ H(t, x)N(dt, dx) |x|<1 + K (t, x)N(dt, dx). |x|≥1Clearly Y is a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 21 / 58
  • 100. Let M = (M(t), t ≥ 0) be an adapted process which is such thatMJ ∈ P2 (t, A) whenever J ∈ P2 (t, A) (where A ∈ B(Rd ) is arbitrary).For example, it is sufficient to take M to be adapted andleft-continuous.For these processes we can define an adapted processZ = (Z (t), t ≥ 0) by the prescription that it have the stochasticdifferential ˜ dZ (t) = M(t)G(t)dt + M(t)F (t)dB(t) + M(t)H(t, x)N(dt, dx) + M(t)K (t, x)N(dt, dx),and we will adopt the natural notation, dZ (t) = M(t)dY (t). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 22 / 58
  • 101. Let M = (M(t), t ≥ 0) be an adapted process which is such thatMJ ∈ P2 (t, A) whenever J ∈ P2 (t, A) (where A ∈ B(Rd ) is arbitrary).For example, it is sufficient to take M to be adapted andleft-continuous.For these processes we can define an adapted processZ = (Z (t), t ≥ 0) by the prescription that it have the stochasticdifferential ˜ dZ (t) = M(t)G(t)dt + M(t)F (t)dB(t) + M(t)H(t, x)N(dt, dx) + M(t)K (t, x)N(dt, dx),and we will adopt the natural notation, dZ (t) = M(t)dY (t). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 22 / 58
  • 102. Let M = (M(t), t ≥ 0) be an adapted process which is such thatMJ ∈ P2 (t, A) whenever J ∈ P2 (t, A) (where A ∈ B(Rd ) is arbitrary).For example, it is sufficient to take M to be adapted andleft-continuous.For these processes we can define an adapted processZ = (Z (t), t ≥ 0) by the prescription that it have the stochasticdifferential ˜ dZ (t) = M(t)G(t)dt + M(t)F (t)dB(t) + M(t)H(t, x)N(dt, dx) + M(t)K (t, x)N(dt, dx),and we will adopt the natural notation, dZ (t) = M(t)dY (t). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 22 / 58
  • 103. Let M = (M(t), t ≥ 0) be an adapted process which is such thatMJ ∈ P2 (t, A) whenever J ∈ P2 (t, A) (where A ∈ B(Rd ) is arbitrary).For example, it is sufficient to take M to be adapted andleft-continuous.For these processes we can define an adapted processZ = (Z (t), t ≥ 0) by the prescription that it have the stochasticdifferential ˜ dZ (t) = M(t)G(t)dt + M(t)F (t)dB(t) + M(t)H(t, x)N(dt, dx) + M(t)K (t, x)N(dt, dx),and we will adopt the natural notation, dZ (t) = M(t)dY (t). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 22 / 58
  • 104. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 105. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 106. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 107. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 108. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 109. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 110. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 111. Example (Lévy Stochastic Integrals)Let X be a Lévy process with characteristics (b, a, ν) and Lévy-Itôdecomposition: X (t) = bt + Ba (t) + ˜ x N(t, dx) + xN(t, dx), |x|<1 |x|≥1for each t ≥ 0. Let L ∈ P2 (t) for all t ≥ 0 and choose eachFji = aji L, H i = K i = x i L. Then we can construct processes with thestochastic differential dY (t) = L(t)dX (t) (0.3)We call Y a Lévy stochastic integral.In the case where X has finite variation, the Lévy stochastic integral Ycan also be constructed as a Lebesgue-Stieltjes integral, and thiscoincides (up to a set of measure zero) with the prescription (0.3). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 23 / 58
  • 112. Example: The Ornstein Uhlenbeck Process (OU Process) t Y (t) = e−λt y0 + e−λ(t−s) dX (s) (0.4) 0where y0 ∈ Rd is fixed, is a Markov process. The condition log(1 + |x|)ν(dx) < ∞ |x|>1is necessary and sufficient for it to be stationary. There are importantapplications to volatility modelling in finance which have beendeveloped by Ole Barndorff-Nielsen and Neil Sheppard. Intriguingly,every self-decomposable random variable can be naturally embeddedin a stationary Lévy-driven OU process.See lecture 5 for more details. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 24 / 58
  • 113. Example: The Ornstein Uhlenbeck Process (OU Process) t Y (t) = e−λt y0 + e−λ(t−s) dX (s) (0.4) 0where y0 ∈ Rd is fixed, is a Markov process. The condition log(1 + |x|)ν(dx) < ∞ |x|>1is necessary and sufficient for it to be stationary. There are importantapplications to volatility modelling in finance which have beendeveloped by Ole Barndorff-Nielsen and Neil Sheppard. Intriguingly,every self-decomposable random variable can be naturally embeddedin a stationary Lévy-driven OU process.See lecture 5 for more details. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 24 / 58
  • 114. Example: The Ornstein Uhlenbeck Process (OU Process) t Y (t) = e−λt y0 + e−λ(t−s) dX (s) (0.4) 0where y0 ∈ Rd is fixed, is a Markov process. The condition log(1 + |x|)ν(dx) < ∞ |x|>1is necessary and sufficient for it to be stationary. There are importantapplications to volatility modelling in finance which have beendeveloped by Ole Barndorff-Nielsen and Neil Sheppard. Intriguingly,every self-decomposable random variable can be naturally embeddedin a stationary Lévy-driven OU process.See lecture 5 for more details. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 24 / 58
  • 115. Example: The Ornstein Uhlenbeck Process (OU Process) t Y (t) = e−λt y0 + e−λ(t−s) dX (s) (0.4) 0where y0 ∈ Rd is fixed, is a Markov process. The condition log(1 + |x|)ν(dx) < ∞ |x|>1is necessary and sufficient for it to be stationary. There are importantapplications to volatility modelling in finance which have beendeveloped by Ole Barndorff-Nielsen and Neil Sheppard. Intriguingly,every self-decomposable random variable can be naturally embeddedin a stationary Lévy-driven OU process.See lecture 5 for more details. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 24 / 58
  • 116. Example: The Ornstein Uhlenbeck Process (OU Process) t Y (t) = e−λt y0 + e−λ(t−s) dX (s) (0.4) 0where y0 ∈ Rd is fixed, is a Markov process. The condition log(1 + |x|)ν(dx) < ∞ |x|>1is necessary and sufficient for it to be stationary. There are importantapplications to volatility modelling in finance which have beendeveloped by Ole Barndorff-Nielsen and Neil Sheppard. Intriguingly,every self-decomposable random variable can be naturally embeddedin a stationary Lévy-driven OU process.See lecture 5 for more details. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 24 / 58
  • 117. Example: The Ornstein Uhlenbeck Process (OU Process) t Y (t) = e−λt y0 + e−λ(t−s) dX (s) (0.4) 0where y0 ∈ Rd is fixed, is a Markov process. The condition log(1 + |x|)ν(dx) < ∞ |x|>1is necessary and sufficient for it to be stationary. There are importantapplications to volatility modelling in finance which have beendeveloped by Ole Barndorff-Nielsen and Neil Sheppard. Intriguingly,every self-decomposable random variable can be naturally embeddedin a stationary Lévy-driven OU process.See lecture 5 for more details. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 24 / 58
  • 118. Stochastic integration has a plethora of applications including filtering,stochastic control and infinite dimensional analysis. Here’s somemotivation from finance. Suppose that X (t) is the value of a stock attime t and F (t) is the number of stocks owned at time t. Assume fornow that we buy and sell stocks at discrete times0 = t0 < t1 < · · · < tn < tn+1 = T . Then the total value of the portfolioat time T is: n V (T ) = V (0) + F (tj )(X (tj+1 ) − X (tj )), j=0and in the limit as the times become infinitesimally close together, wehave the stochastic integral t V (T ) = V (0) + F (s)dX (s). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 25 / 58
  • 119. Stochastic integration has a plethora of applications including filtering,stochastic control and infinite dimensional analysis. Here’s somemotivation from finance. Suppose that X (t) is the value of a stock attime t and F (t) is the number of stocks owned at time t. Assume fornow that we buy and sell stocks at discrete times0 = t0 < t1 < · · · < tn < tn+1 = T . Then the total value of the portfolioat time T is: n V (T ) = V (0) + F (tj )(X (tj+1 ) − X (tj )), j=0and in the limit as the times become infinitesimally close together, wehave the stochastic integral t V (T ) = V (0) + F (s)dX (s). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 25 / 58
  • 120. Stochastic integration has a plethora of applications including filtering,stochastic control and infinite dimensional analysis. Here’s somemotivation from finance. Suppose that X (t) is the value of a stock attime t and F (t) is the number of stocks owned at time t. Assume fornow that we buy and sell stocks at discrete times0 = t0 < t1 < · · · < tn < tn+1 = T . Then the total value of the portfolioat time T is: n V (T ) = V (0) + F (tj )(X (tj+1 ) − X (tj )), j=0and in the limit as the times become infinitesimally close together, wehave the stochastic integral t V (T ) = V (0) + F (s)dX (s). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 25 / 58
  • 121. Stochastic integration has a plethora of applications including filtering,stochastic control and infinite dimensional analysis. Here’s somemotivation from finance. Suppose that X (t) is the value of a stock attime t and F (t) is the number of stocks owned at time t. Assume fornow that we buy and sell stocks at discrete times0 = t0 < t1 < · · · < tn < tn+1 = T . Then the total value of the portfolioat time T is: n V (T ) = V (0) + F (tj )(X (tj+1 ) − X (tj )), j=0and in the limit as the times become infinitesimally close together, wehave the stochastic integral t V (T ) = V (0) + F (s)dX (s). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 25 / 58
  • 122. Stochastic integration against Brownian motion was first developed byWiener for sure functions. Itô’s groundbreaking work in 1944 extendedthis to random adapted integrands.The generalisation of the integratorto arbitrary martingales was due to Kunita and Watanabe in 1967andthe further step to allow semimartingales was due to P.A.Meyer andthe Strasbourg school in the 1970s. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 26 / 58
  • 123. Stochastic integration against Brownian motion was first developed byWiener for sure functions. Itô’s groundbreaking work in 1944 extendedthis to random adapted integrands.The generalisation of the integratorto arbitrary martingales was due to Kunita and Watanabe in 1967andthe further step to allow semimartingales was due to P.A.Meyer andthe Strasbourg school in the 1970s. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 26 / 58
  • 124. Stochastic integration against Brownian motion was first developed byWiener for sure functions. Itô’s groundbreaking work in 1944 extendedthis to random adapted integrands.The generalisation of the integratorto arbitrary martingales was due to Kunita and Watanabe in 1967andthe further step to allow semimartingales was due to P.A.Meyer andthe Strasbourg school in the 1970s. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 26 / 58
  • 125. Stochastic integration against Brownian motion was first developed byWiener for sure functions. Itô’s groundbreaking work in 1944 extendedthis to random adapted integrands.The generalisation of the integratorto arbitrary martingales was due to Kunita and Watanabe in 1967andthe further step to allow semimartingales was due to P.A.Meyer andthe Strasbourg school in the 1970s. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 26 / 58
  • 126. Itô’s FormulaWe begin with the easy case - Itô’s formula for Poisson stochasticintegrals of the form t W i (t) = W i (0) + K i (t, x)N(dt, dx) (0.5) 0 Afor 1 ≤ i ≤ d, where t ≥ 0, A is bounded below and each K i ispredictable. Itô’s formula for such processes takes a particularly simpleform. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 27 / 58
  • 127. Itô’s FormulaWe begin with the easy case - Itô’s formula for Poisson stochasticintegrals of the form t W i (t) = W i (0) + K i (t, x)N(dt, dx) (0.5) 0 Afor 1 ≤ i ≤ d, where t ≥ 0, A is bounded below and each K i ispredictable. Itô’s formula for such processes takes a particularly simpleform. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 27 / 58
  • 128. Itô’s FormulaWe begin with the easy case - Itô’s formula for Poisson stochasticintegrals of the form t W i (t) = W i (0) + K i (t, x)N(dt, dx) (0.5) 0 Afor 1 ≤ i ≤ d, where t ≥ 0, A is bounded below and each K i ispredictable. Itô’s formula for such processes takes a particularly simpleform. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 27 / 58
  • 129. LemmaIf W is a Poisson stochastic integral of the above form then for eachf ∈ C(Rd ), and for each t ≥ 0, with probability one, we have tf (W (t)) − f (W (0)) = [f (W (s−) + K (s, x)) − f (W (s−))]N(ds, dx). 0 A Dave Applebaum (Sheffield UK) Lecture 4 December 2011 28 / 58
  • 130. Proof. Let Y (t) = A xN(dt, dx) and recall that the jump times for Y are Adefined recursively as T0 = 0 and for each A An ∈ N, Tn = inf{t > Tn−1 ; ∆Y (t) ∈ A}. We then find that, f ((W (t)) − f (W (0)) = f (W (s)) − f (W (s−)) 0≤s≤t ∞ A A = f (W (t ∧ Tn )) − f (W (t ∧ Tn−1 )) n=1 ∞ A A A A = [f (W (t ∧ Tn −) + K (t ∧ Tn , ∆Y (t ∧ Tn ))) − f (W (t ∧ Tn −))] n=1 t = [f (W (s−) + K (s, x)) − f (W (s−))]N(ds, dx). 2 0 A Dave Applebaum (Sheffield UK) Lecture 4 December 2011 29 / 58
  • 131. Proof. Let Y (t) = A xN(dt, dx) and recall that the jump times for Y are Adefined recursively as T0 = 0 and for each A An ∈ N, Tn = inf{t > Tn−1 ; ∆Y (t) ∈ A}. We then find that, f ((W (t)) − f (W (0)) = f (W (s)) − f (W (s−)) 0≤s≤t ∞ A A = f (W (t ∧ Tn )) − f (W (t ∧ Tn−1 )) n=1 ∞ A A A A = [f (W (t ∧ Tn −) + K (t ∧ Tn , ∆Y (t ∧ Tn ))) − f (W (t ∧ Tn −))] n=1 t = [f (W (s−) + K (s, x)) − f (W (s−))]N(ds, dx). 2 0 A Dave Applebaum (Sheffield UK) Lecture 4 December 2011 29 / 58
  • 132. Proof. Let Y (t) = A xN(dt, dx) and recall that the jump times for Y are Adefined recursively as T0 = 0 and for each A An ∈ N, Tn = inf{t > Tn−1 ; ∆Y (t) ∈ A}. We then find that, f ((W (t)) − f (W (0)) = f (W (s)) − f (W (s−)) 0≤s≤t ∞ A A = f (W (t ∧ Tn )) − f (W (t ∧ Tn−1 )) n=1 ∞ A A A A = [f (W (t ∧ Tn −) + K (t ∧ Tn , ∆Y (t ∧ Tn ))) − f (W (t ∧ Tn −))] n=1 t = [f (W (s−) + K (s, x)) − f (W (s−))]N(ds, dx). 2 0 A Dave Applebaum (Sheffield UK) Lecture 4 December 2011 29 / 58
  • 133. Proof. Let Y (t) = A xN(dt, dx) and recall that the jump times for Y are Adefined recursively as T0 = 0 and for each A An ∈ N, Tn = inf{t > Tn−1 ; ∆Y (t) ∈ A}. We then find that, f ((W (t)) − f (W (0)) = f (W (s)) − f (W (s−)) 0≤s≤t ∞ A A = f (W (t ∧ Tn )) − f (W (t ∧ Tn−1 )) n=1 ∞ A A A A = [f (W (t ∧ Tn −) + K (t ∧ Tn , ∆Y (t ∧ Tn ))) − f (W (t ∧ Tn −))] n=1 t = [f (W (s−) + K (s, x)) − f (W (s−))]N(ds, dx). 2 0 A Dave Applebaum (Sheffield UK) Lecture 4 December 2011 29 / 58
  • 134. Proof. Let Y (t) = A xN(dt, dx) and recall that the jump times for Y are Adefined recursively as T0 = 0 and for each A An ∈ N, Tn = inf{t > Tn−1 ; ∆Y (t) ∈ A}. We then find that, f ((W (t)) − f (W (0)) = f (W (s)) − f (W (s−)) 0≤s≤t ∞ A A = f (W (t ∧ Tn )) − f (W (t ∧ Tn−1 )) n=1 ∞ A A A A = [f (W (t ∧ Tn −) + K (t ∧ Tn , ∆Y (t ∧ Tn ))) − f (W (t ∧ Tn −))] n=1 t = [f (W (s−) + K (s, x)) − f (W (s−))]N(ds, dx). 2 0 A Dave Applebaum (Sheffield UK) Lecture 4 December 2011 29 / 58
  • 135. The celebrated Itô formula for Brownian motion is probably well-knownto you so I’ll briefly outline the proof. Let (Pn , n ∈ N) be a sequence ofpartitions of the form (n) (n) (n) (n)Pn = {0 = t0 < t1 < . . . < tm(n) < tm(n)+1 = T }. and suppose that (n) (n)limn→∞ δ(Pn ) = 0, where the mesh, δ(Pn ) = max0≤j≤m(n) |tj+1 − tj |.As a preliminary - you need the following:-LemmaIf Wkl ∈ H2 (T ) for each 1 ≤ k , l ≤ m, then n (n) (n) (n) (n) (n) L2 − lim Wkl (tj )(B k (tj+1 ) − B k (tj ))(B l (tj+1 ) − B l (tj )) n→∞ j=0 m T = Wkk (s)ds. k =1 0The proof is similar to that of Lemma 1 - but you will need theGaussian moment E(B(t)4 ) = 3t 2 . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 30 / 58
  • 136. The celebrated Itô formula for Brownian motion is probably well-knownto you so I’ll briefly outline the proof. Let (Pn , n ∈ N) be a sequence ofpartitions of the form (n) (n) (n) (n)Pn = {0 = t0 < t1 < . . . < tm(n) < tm(n)+1 = T }. and suppose that (n) (n)limn→∞ δ(Pn ) = 0, where the mesh, δ(Pn ) = max0≤j≤m(n) |tj+1 − tj |.As a preliminary - you need the following:-LemmaIf Wkl ∈ H2 (T ) for each 1 ≤ k , l ≤ m, then n (n) (n) (n) (n) (n) L2 − lim Wkl (tj )(B k (tj+1 ) − B k (tj ))(B l (tj+1 ) − B l (tj )) n→∞ j=0 m T = Wkk (s)ds. k =1 0The proof is similar to that of Lemma 1 - but you will need theGaussian moment E(B(t)4 ) = 3t 2 . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 30 / 58
  • 137. The celebrated Itô formula for Brownian motion is probably well-knownto you so I’ll briefly outline the proof. Let (Pn , n ∈ N) be a sequence ofpartitions of the form (n) (n) (n) (n)Pn = {0 = t0 < t1 < . . . < tm(n) < tm(n)+1 = T }. and suppose that (n) (n)limn→∞ δ(Pn ) = 0, where the mesh, δ(Pn ) = max0≤j≤m(n) |tj+1 − tj |.As a preliminary - you need the following:-LemmaIf Wkl ∈ H2 (T ) for each 1 ≤ k , l ≤ m, then n (n) (n) (n) (n) (n) L2 − lim Wkl (tj )(B k (tj+1 ) − B k (tj ))(B l (tj+1 ) − B l (tj )) n→∞ j=0 m T = Wkk (s)ds. k =1 0The proof is similar to that of Lemma 1 - but you will need theGaussian moment E(B(t)4 ) = 3t 2 . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 30 / 58
  • 138. The celebrated Itô formula for Brownian motion is probably well-knownto you so I’ll briefly outline the proof. Let (Pn , n ∈ N) be a sequence ofpartitions of the form (n) (n) (n) (n)Pn = {0 = t0 < t1 < . . . < tm(n) < tm(n)+1 = T }. and suppose that (n) (n)limn→∞ δ(Pn ) = 0, where the mesh, δ(Pn ) = max0≤j≤m(n) |tj+1 − tj |.As a preliminary - you need the following:-LemmaIf Wkl ∈ H2 (T ) for each 1 ≤ k , l ≤ m, then n (n) (n) (n) (n) (n) L2 − lim Wkl (tj )(B k (tj+1 ) − B k (tj ))(B l (tj+1 ) − B l (tj )) n→∞ j=0 m T = Wkk (s)ds. k =1 0The proof is similar to that of Lemma 1 - but you will need theGaussian moment E(B(t)4 ) = 3t 2 . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 30 / 58
  • 139. The celebrated Itô formula for Brownian motion is probably well-knownto you so I’ll briefly outline the proof. Let (Pn , n ∈ N) be a sequence ofpartitions of the form (n) (n) (n) (n)Pn = {0 = t0 < t1 < . . . < tm(n) < tm(n)+1 = T }. and suppose that (n) (n)limn→∞ δ(Pn ) = 0, where the mesh, δ(Pn ) = max0≤j≤m(n) |tj+1 − tj |.As a preliminary - you need the following:-LemmaIf Wkl ∈ H2 (T ) for each 1 ≤ k , l ≤ m, then n (n) (n) (n) (n) (n) L2 − lim Wkl (tj )(B k (tj+1 ) − B k (tj ))(B l (tj+1 ) − B l (tj )) n→∞ j=0 m T = Wkk (s)ds. k =1 0The proof is similar to that of Lemma 1 - but you will need theGaussian moment E(B(t)4 ) = 3t 2 . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 30 / 58
  • 140. Now let M be a Brownian integral with drift of the form t t M i (t) = Fji (s)dB j (s) + Gi (s)ds, (0.6) 0 0 1where each Fji , (Gi ) 2 ∈ P2 (t), for all t ≥ 0, 1 ≤ i ≤ d, 1 ≤ j ≤ m.For each 1 ≤ i ≤ j, we introduce the quadratic variation processdenoted as ([M i , M j ](t), t ≥ 0) by m T j [M i , M j ](t) = i Fk (s)Fk (s)ds. k =1 0We will explore quadratic variation in greater depth in the sequel. Thefollowing slick method of proving Itô’s formula is due to Kunita. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 31 / 58
  • 141. Now let M be a Brownian integral with drift of the form t t M i (t) = Fji (s)dB j (s) + Gi (s)ds, (0.6) 0 0 1where each Fji , (Gi ) 2 ∈ P2 (t), for all t ≥ 0, 1 ≤ i ≤ d, 1 ≤ j ≤ m.For each 1 ≤ i ≤ j, we introduce the quadratic variation processdenoted as ([M i , M j ](t), t ≥ 0) by m T j [M i , M j ](t) = i Fk (s)Fk (s)ds. k =1 0We will explore quadratic variation in greater depth in the sequel. Thefollowing slick method of proving Itô’s formula is due to Kunita. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 31 / 58
  • 142. Now let M be a Brownian integral with drift of the form t t M i (t) = Fji (s)dB j (s) + Gi (s)ds, (0.6) 0 0 1where each Fji , (Gi ) 2 ∈ P2 (t), for all t ≥ 0, 1 ≤ i ≤ d, 1 ≤ j ≤ m.For each 1 ≤ i ≤ j, we introduce the quadratic variation processdenoted as ([M i , M j ](t), t ≥ 0) by m T j [M i , M j ](t) = i Fk (s)Fk (s)ds. k =1 0We will explore quadratic variation in greater depth in the sequel. Thefollowing slick method of proving Itô’s formula is due to Kunita. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 31 / 58
  • 143. Now let M be a Brownian integral with drift of the form t t M i (t) = Fji (s)dB j (s) + Gi (s)ds, (0.6) 0 0 1where each Fji , (Gi ) 2 ∈ P2 (t), for all t ≥ 0, 1 ≤ i ≤ d, 1 ≤ j ≤ m.For each 1 ≤ i ≤ j, we introduce the quadratic variation processdenoted as ([M i , M j ](t), t ≥ 0) by m T j [M i , M j ](t) = i Fk (s)Fk (s)ds. k =1 0We will explore quadratic variation in greater depth in the sequel. Thefollowing slick method of proving Itô’s formula is due to Kunita. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 31 / 58
  • 144. Now let M be a Brownian integral with drift of the form t t M i (t) = Fji (s)dB j (s) + Gi (s)ds, (0.6) 0 0 1where each Fji , (Gi ) 2 ∈ P2 (t), for all t ≥ 0, 1 ≤ i ≤ d, 1 ≤ j ≤ m.For each 1 ≤ i ≤ j, we introduce the quadratic variation processdenoted as ([M i , M j ](t), t ≥ 0) by m T j [M i , M j ](t) = i Fk (s)Fk (s)ds. k =1 0We will explore quadratic variation in greater depth in the sequel. Thefollowing slick method of proving Itô’s formula is due to Kunita. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 31 / 58
  • 145. Now let M be a Brownian integral with drift of the form t t M i (t) = Fji (s)dB j (s) + Gi (s)ds, (0.6) 0 0 1where each Fji , (Gi ) 2 ∈ P2 (t), for all t ≥ 0, 1 ≤ i ≤ d, 1 ≤ j ≤ m.For each 1 ≤ i ≤ j, we introduce the quadratic variation processdenoted as ([M i , M j ](t), t ≥ 0) by m T j [M i , M j ](t) = i Fk (s)Fk (s)ds. k =1 0We will explore quadratic variation in greater depth in the sequel. Thefollowing slick method of proving Itô’s formula is due to Kunita. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 31 / 58
  • 146. Theorem (Itô’s Theorem 1)If M = (M(t), t ≥ 0) is a Brownian integral with drift of the form (0.6),then for all f ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have t t 1f (M(t))−f (M(0)) = ∂i f (M(s))dM i (s)+ ∂i ∂j f (M(s))d[M i , M j ](s). 0 2 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 32 / 58
  • 147. Theorem (Itô’s Theorem 1)If M = (M(t), t ≥ 0) is a Brownian integral with drift of the form (0.6),then for all f ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have t t 1f (M(t))−f (M(0)) = ∂i f (M(s))dM i (s)+ ∂i ∂j f (M(s))d[M i , M j ](s). 0 2 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 32 / 58
  • 148. Theorem (Itô’s Theorem 1)If M = (M(t), t ≥ 0) is a Brownian integral with drift of the form (0.6),then for all f ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have t t 1f (M(t))−f (M(0)) = ∂i f (M(s))dM i (s)+ ∂i ∂j f (M(s))d[M i , M j ](s). 0 2 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 32 / 58
  • 149. Proof. Let (Pn , n ∈ N) be a sequence of partitions of [0, t] as above.By Taylor’s theorem, we have, for each such partition (where wesuppress the index n). m f (M(t)) − f (M(0)) = f (M(tk +1 )) − f (M(tk )) k =0 1 = J1 (t) + J2 (t), 2where m J1 (t) = ∂i f (M(tk ))(M i (tk +1 ) − M i (tk )), k =0 m J2 (t) = ∂i ∂j f (Nij )(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k k =0and where the Nij ’s are each F(tk +1 )-adapted Rd -valued random k kvariables satisfying |Nij − M(tk )| ≤ |M(tk +1 ) − M(tk )|. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 33 / 58
  • 150. Proof. Let (Pn , n ∈ N) be a sequence of partitions of [0, t] as above.By Taylor’s theorem, we have, for each such partition (where wesuppress the index n). m f (M(t)) − f (M(0)) = f (M(tk +1 )) − f (M(tk )) k =0 1 = J1 (t) + J2 (t), 2where m J1 (t) = ∂i f (M(tk ))(M i (tk +1 ) − M i (tk )), k =0 m J2 (t) = ∂i ∂j f (Nij )(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k k =0and where the Nij ’s are each F(tk +1 )-adapted Rd -valued random k kvariables satisfying |Nij − M(tk )| ≤ |M(tk +1 ) − M(tk )|. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 33 / 58
  • 151. Proof. Let (Pn , n ∈ N) be a sequence of partitions of [0, t] as above.By Taylor’s theorem, we have, for each such partition (where wesuppress the index n). m f (M(t)) − f (M(0)) = f (M(tk +1 )) − f (M(tk )) k =0 1 = J1 (t) + J2 (t), 2where m J1 (t) = ∂i f (M(tk ))(M i (tk +1 ) − M i (tk )), k =0 m J2 (t) = ∂i ∂j f (Nij )(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k k =0and where the Nij ’s are each F(tk +1 )-adapted Rd -valued random k kvariables satisfying |Nij − M(tk )| ≤ |M(tk +1 ) − M(tk )|. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 33 / 58
  • 152. Proof. Let (Pn , n ∈ N) be a sequence of partitions of [0, t] as above.By Taylor’s theorem, we have, for each such partition (where wesuppress the index n). m f (M(t)) − f (M(0)) = f (M(tk +1 )) − f (M(tk )) k =0 1 = J1 (t) + J2 (t), 2where m J1 (t) = ∂i f (M(tk ))(M i (tk +1 ) − M i (tk )), k =0 m J2 (t) = ∂i ∂j f (Nij )(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k k =0and where the Nij ’s are each F(tk +1 )-adapted Rd -valued random k kvariables satisfying |Nij − M(tk )| ≤ |M(tk +1 ) − M(tk )|. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 33 / 58
  • 153. Proof. Let (Pn , n ∈ N) be a sequence of partitions of [0, t] as above.By Taylor’s theorem, we have, for each such partition (where wesuppress the index n). m f (M(t)) − f (M(0)) = f (M(tk +1 )) − f (M(tk )) k =0 1 = J1 (t) + J2 (t), 2where m J1 (t) = ∂i f (M(tk ))(M i (tk +1 ) − M i (tk )), k =0 m J2 (t) = ∂i ∂j f (Nij )(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k k =0and where the Nij ’s are each F(tk +1 )-adapted Rd -valued random k kvariables satisfying |Nij − M(tk )| ≤ |M(tk +1 ) − M(tk )|. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 33 / 58
  • 154. Proof. Let (Pn , n ∈ N) be a sequence of partitions of [0, t] as above.By Taylor’s theorem, we have, for each such partition (where wesuppress the index n). m f (M(t)) − f (M(0)) = f (M(tk +1 )) − f (M(tk )) k =0 1 = J1 (t) + J2 (t), 2where m J1 (t) = ∂i f (M(tk ))(M i (tk +1 ) − M i (tk )), k =0 m J2 (t) = ∂i ∂j f (Nij )(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k k =0and where the Nij ’s are each F(tk +1 )-adapted Rd -valued random k kvariables satisfying |Nij − M(tk )| ≤ |M(tk +1 ) − M(tk )|. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 33 / 58
  • 155. We write each J2 (t) = K1 (t) + K2 (t),where m K1 (t) = ∂i ∂j f (M(tk ))(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k =0 mK2 (t) = k [∂i ∂j f (Nij )−∂i ∂j f (M(tk ))](M i (tk +1 )−M i (tk ))(M j (tk +1 )−M j (tk )). k =0Now take limits as n → ∞. It turns out that K2 (t) → 0, in probabilityand the result follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 34 / 58
  • 156. We write each J2 (t) = K1 (t) + K2 (t),where m K1 (t) = ∂i ∂j f (M(tk ))(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k =0 mK2 (t) = k [∂i ∂j f (Nij )−∂i ∂j f (M(tk ))](M i (tk +1 )−M i (tk ))(M j (tk +1 )−M j (tk )). k =0Now take limits as n → ∞. It turns out that K2 (t) → 0, in probabilityand the result follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 34 / 58
  • 157. We write each J2 (t) = K1 (t) + K2 (t),where m K1 (t) = ∂i ∂j f (M(tk ))(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k =0 mK2 (t) = k [∂i ∂j f (Nij )−∂i ∂j f (M(tk ))](M i (tk +1 )−M i (tk ))(M j (tk +1 )−M j (tk )). k =0Now take limits as n → ∞. It turns out that K2 (t) → 0, in probabilityand the result follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 34 / 58
  • 158. We write each J2 (t) = K1 (t) + K2 (t),where m K1 (t) = ∂i ∂j f (M(tk ))(M i (tk +1 ) − M i (tk ))(M j (tk +1 ) − M j (tk )), k =0 mK2 (t) = k [∂i ∂j f (Nij )−∂i ∂j f (M(tk ))](M i (tk +1 )−M i (tk ))(M j (tk +1 )−M j (tk )). k =0Now take limits as n → ∞. It turns out that K2 (t) → 0, in probabilityand the result follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 34 / 58
  • 159. Itô’s formula for general Lévy-type stochastic integrals is obtainedessentially by combining the Poisson and Brownian results and makingsure you take good care of the compensators for small jumps. Youshould be able to guess the right result.To give a precise statement, consider a Lévy-type stochastic integral ofthe form ˜ dY (t) = G(t)dt +F (t)dB(t)+H(t, x)N(dt, dx)+K (t, x)N(dt, dx). (0.7) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 35 / 58
  • 160. Itô’s formula for general Lévy-type stochastic integrals is obtainedessentially by combining the Poisson and Brownian results and makingsure you take good care of the compensators for small jumps. Youshould be able to guess the right result.To give a precise statement, consider a Lévy-type stochastic integral ofthe form ˜ dY (t) = G(t)dt +F (t)dB(t)+H(t, x)N(dt, dx)+K (t, x)N(dt, dx). (0.7) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 35 / 58
  • 161. Itô’s formula for general Lévy-type stochastic integrals is obtainedessentially by combining the Poisson and Brownian results and makingsure you take good care of the compensators for small jumps. Youshould be able to guess the right result.To give a precise statement, consider a Lévy-type stochastic integral ofthe form ˜ dY (t) = G(t)dt +F (t)dB(t)+H(t, x)N(dt, dx)+K (t, x)N(dt, dx). (0.7) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 35 / 58
  • 162. Itô’s formula for general Lévy-type stochastic integrals is obtainedessentially by combining the Poisson and Brownian results and makingsure you take good care of the compensators for small jumps. Youshould be able to guess the right result.To give a precise statement, consider a Lévy-type stochastic integral ofthe form ˜ dY (t) = G(t)dt +F (t)dB(t)+H(t, x)N(dt, dx)+K (t, x)N(dt, dx). (0.7) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 35 / 58
  • 163. Theorem (Itô’s Theorem 2)If Y is a Lévy-type stochastic integral of the above form then for eachf ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have f (Y (t)) − f (Y (0)) t t i 1 i j = ∂i f (Y (s−))dYc (s) + ∂i ∂j f (Y (s−))d[Yc , Yc ](s) 0 2 0 t + [f (Y (s−) + K (s, x)) − f (Y (s−))]N(ds, dx) 0 |x|≥1 t + ˜ [f (Y (s−) + H(s, x)) − f (Y (s−))]N(ds, dx) 0 |x|<1 t + [f (Y (s−) + H(s, x)) − f (Y (s−)) 0 |x|<1 − H i (s, x)∂i f (Y (s−)) ν(dx)ds. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 36 / 58
  • 164. Theorem (Itô’s Theorem 2)If Y is a Lévy-type stochastic integral of the above form then for eachf ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have f (Y (t)) − f (Y (0)) t t i 1 i j = ∂i f (Y (s−))dYc (s) + ∂i ∂j f (Y (s−))d[Yc , Yc ](s) 0 2 0 t + [f (Y (s−) + K (s, x)) − f (Y (s−))]N(ds, dx) 0 |x|≥1 t + ˜ [f (Y (s−) + H(s, x)) − f (Y (s−))]N(ds, dx) 0 |x|<1 t + [f (Y (s−) + H(s, x)) − f (Y (s−)) 0 |x|<1 − H i (s, x)∂i f (Y (s−)) ν(dx)ds. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 36 / 58
  • 165. Here Yc denotes the continuous part of Y defined by t tYc (t) = 0 Gi (s)ds + 0 Fji (s)dB j (s). iTedious but straightforward algebra yields the following form, which isimportant since it extends to general semimartingales:-Theorem (Itô’s Theorem 3)If Y is a Lévy-type stochastic integral of the above form then for eachf ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have f (Y (t)) − f (Y (0)) t t 1 j = ∂i f (Y (s−))dY i (s) + i ∂i ∂j f (Y (s−))d[Yc , Yc ](s) 0 2 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 37 / 58
  • 166. Here Yc denotes the continuous part of Y defined by t tYc (t) = 0 Gi (s)ds + 0 Fji (s)dB j (s). iTedious but straightforward algebra yields the following form, which isimportant since it extends to general semimartingales:-Theorem (Itô’s Theorem 3)If Y is a Lévy-type stochastic integral of the above form then for eachf ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have f (Y (t)) − f (Y (0)) t t 1 j = ∂i f (Y (s−))dY i (s) + i ∂i ∂j f (Y (s−))d[Yc , Yc ](s) 0 2 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 37 / 58
  • 167. Here Yc denotes the continuous part of Y defined by t tYc (t) = 0 Gi (s)ds + 0 Fji (s)dB j (s). iTedious but straightforward algebra yields the following form, which isimportant since it extends to general semimartingales:-Theorem (Itô’s Theorem 3)If Y is a Lévy-type stochastic integral of the above form then for eachf ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have f (Y (t)) − f (Y (0)) t t 1 j = ∂i f (Y (s−))dY i (s) + i ∂i ∂j f (Y (s−))d[Yc , Yc ](s) 0 2 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 37 / 58
  • 168. Here Yc denotes the continuous part of Y defined by t tYc (t) = 0 Gi (s)ds + 0 Fji (s)dB j (s). iTedious but straightforward algebra yields the following form, which isimportant since it extends to general semimartingales:-Theorem (Itô’s Theorem 3)If Y is a Lévy-type stochastic integral of the above form then for eachf ∈ C 2 (Rd ), t ≥ 0, with probability 1, we have f (Y (t)) − f (Y (0)) t t 1 j = ∂i f (Y (s−))dY i (s) + i ∂i ∂j f (Y (s−))d[Yc , Yc ](s) 0 2 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤t Dave Applebaum (Sheffield UK) Lecture 4 December 2011 37 / 58
  • 169. Note that a special case of Itô’s formula yields the following “classical”chain rule for differentiable functions f , when the process Y is of finitevariation: tf (Y (t)) − f (Y (0)) = ∂i f (Y (s−))dY i (s) + 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤tA form of Itô’s formula may even be established for fractional Brownianmotion, which is not a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 38 / 58
  • 170. Note that a special case of Itô’s formula yields the following “classical”chain rule for differentiable functions f , when the process Y is of finitevariation: tf (Y (t)) − f (Y (0)) = ∂i f (Y (s−))dY i (s) + 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤tA form of Itô’s formula may even be established for fractional Brownianmotion, which is not a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 38 / 58
  • 171. Note that a special case of Itô’s formula yields the following “classical”chain rule for differentiable functions f , when the process Y is of finitevariation: tf (Y (t)) − f (Y (0)) = ∂i f (Y (s−))dY i (s) + 0 + [f (Y (s)) − f (Y (s−)) − ∆Y i (s)∂i f (Y (s−))]. 0≤s≤tA form of Itô’s formula may even be established for fractional Brownianmotion, which is not a semimartingale. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 38 / 58
  • 172. Quadratic Variation and Itô’s Product FormulaWe extend the definition of quadratic variation to the more generalcase of Lévy-type stochastic integrals Y = (Y (t), t ≥ 0). So for eacht ≥ 0 we define a d × d matrix-valued adapted process[Y , Y ] = ([Y , Y ](t), t ≥ 0) by the following prescription for its (i, j)thentry (1 ≤ i, j ≤ d), j [Y i , Y j ](t) = [Yc , Yc ](t) + i ∆Y i (s)∆Y j (s). (0.8) 0≤s≤tEach [Y i , Y j ](t) is almost surely finite, and we have m T t j[Y i , Y j ](t) = i Fk (s)Fk (s)ds + H i (s, x)H j (s, x)N(ds, dx) k =1 0 0 |x|<1 t + K i (s, x)K j (s, x)N(ds, dx), (0.9 0 |x|≥1so that we clearly have each [Y i , Y j ](t) = [Y j , Y i ](t). Note that theintegral over small jumps in this case is always a.s. finite (Why ?) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 39 / 58
  • 173. Quadratic Variation and Itô’s Product FormulaWe extend the definition of quadratic variation to the more generalcase of Lévy-type stochastic integrals Y = (Y (t), t ≥ 0). So for eacht ≥ 0 we define a d × d matrix-valued adapted process[Y , Y ] = ([Y , Y ](t), t ≥ 0) by the following prescription for its (i, j)thentry (1 ≤ i, j ≤ d), j [Y i , Y j ](t) = [Yc , Yc ](t) + i ∆Y i (s)∆Y j (s). (0.8) 0≤s≤tEach [Y i , Y j ](t) is almost surely finite, and we have m T t j[Y i , Y j ](t) = i Fk (s)Fk (s)ds + H i (s, x)H j (s, x)N(ds, dx) k =1 0 0 |x|<1 t + K i (s, x)K j (s, x)N(ds, dx), (0.9 0 |x|≥1so that we clearly have each [Y i , Y j ](t) = [Y j , Y i ](t). Note that theintegral over small jumps in this case is always a.s. finite (Why ?) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 39 / 58
  • 174. Quadratic Variation and Itô’s Product FormulaWe extend the definition of quadratic variation to the more generalcase of Lévy-type stochastic integrals Y = (Y (t), t ≥ 0). So for eacht ≥ 0 we define a d × d matrix-valued adapted process[Y , Y ] = ([Y , Y ](t), t ≥ 0) by the following prescription for its (i, j)thentry (1 ≤ i, j ≤ d), j [Y i , Y j ](t) = [Yc , Yc ](t) + i ∆Y i (s)∆Y j (s). (0.8) 0≤s≤tEach [Y i , Y j ](t) is almost surely finite, and we have m T t j[Y i , Y j ](t) = i Fk (s)Fk (s)ds + H i (s, x)H j (s, x)N(ds, dx) k =1 0 0 |x|<1 t + K i (s, x)K j (s, x)N(ds, dx), (0.9 0 |x|≥1so that we clearly have each [Y i , Y j ](t) = [Y j , Y i ](t). Note that theintegral over small jumps in this case is always a.s. finite (Why ?) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 39 / 58
  • 175. Quadratic Variation and Itô’s Product FormulaWe extend the definition of quadratic variation to the more generalcase of Lévy-type stochastic integrals Y = (Y (t), t ≥ 0). So for eacht ≥ 0 we define a d × d matrix-valued adapted process[Y , Y ] = ([Y , Y ](t), t ≥ 0) by the following prescription for its (i, j)thentry (1 ≤ i, j ≤ d), j [Y i , Y j ](t) = [Yc , Yc ](t) + i ∆Y i (s)∆Y j (s). (0.8) 0≤s≤tEach [Y i , Y j ](t) is almost surely finite, and we have m T t j[Y i , Y j ](t) = i Fk (s)Fk (s)ds + H i (s, x)H j (s, x)N(ds, dx) k =1 0 0 |x|<1 t + K i (s, x)K j (s, x)N(ds, dx), (0.9 0 |x|≥1so that we clearly have each [Y i , Y j ](t) = [Y j , Y i ](t). Note that theintegral over small jumps in this case is always a.s. finite (Why ?) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 39 / 58
  • 176. Quadratic Variation and Itô’s Product FormulaWe extend the definition of quadratic variation to the more generalcase of Lévy-type stochastic integrals Y = (Y (t), t ≥ 0). So for eacht ≥ 0 we define a d × d matrix-valued adapted process[Y , Y ] = ([Y , Y ](t), t ≥ 0) by the following prescription for its (i, j)thentry (1 ≤ i, j ≤ d), j [Y i , Y j ](t) = [Yc , Yc ](t) + i ∆Y i (s)∆Y j (s). (0.8) 0≤s≤tEach [Y i , Y j ](t) is almost surely finite, and we have m T t j[Y i , Y j ](t) = i Fk (s)Fk (s)ds + H i (s, x)H j (s, x)N(ds, dx) k =1 0 0 |x|<1 t + K i (s, x)K j (s, x)N(ds, dx), (0.9 0 |x|≥1so that we clearly have each [Y i , Y j ](t) = [Y j , Y i ](t). Note that theintegral over small jumps in this case is always a.s. finite (Why ?) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 39 / 58
  • 177. It is easy to show that for each α, β ∈ R and 1 ≤ i, j, k ≤ d, t ≥ 0, [αY i + βY j , Y k ](t) = α[Y i , Y k ](t) + β[Y j , Y k ](t).The importance of [Y , Y ] is that it measures the deviation in thestochastic differential of products from the usual Leibniz formula. Thefollowing result makes this preciseTheorem (Itô’s Product Formula)If Y 1 and Y 2 are real-valued Lévy-type stochastic integrals of the form(0.7), then for all t ≥ 0, with probability one, we have that t Y 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Y 1 , Y 2 ](t). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 40 / 58
  • 178. It is easy to show that for each α, β ∈ R and 1 ≤ i, j, k ≤ d, t ≥ 0, [αY i + βY j , Y k ](t) = α[Y i , Y k ](t) + β[Y j , Y k ](t).The importance of [Y , Y ] is that it measures the deviation in thestochastic differential of products from the usual Leibniz formula. Thefollowing result makes this preciseTheorem (Itô’s Product Formula)If Y 1 and Y 2 are real-valued Lévy-type stochastic integrals of the form(0.7), then for all t ≥ 0, with probability one, we have that t Y 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Y 1 , Y 2 ](t). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 40 / 58
  • 179. It is easy to show that for each α, β ∈ R and 1 ≤ i, j, k ≤ d, t ≥ 0, [αY i + βY j , Y k ](t) = α[Y i , Y k ](t) + β[Y j , Y k ](t).The importance of [Y , Y ] is that it measures the deviation in thestochastic differential of products from the usual Leibniz formula. Thefollowing result makes this preciseTheorem (Itô’s Product Formula)If Y 1 and Y 2 are real-valued Lévy-type stochastic integrals of the form(0.7), then for all t ≥ 0, with probability one, we have that t Y 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Y 1 , Y 2 ](t). 0 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 40 / 58
  • 180. Proof. We consider Y 1 and Y 2 as components of a vectorY = (Y 1 , Y 2 ) and we take f in Theorem 7 to be the smooth mappingfrom R2 to R given by f (x 1 , x 2 ) = x 1 x 2 .By Theorem 7, we then obtain, for each t ≥ 0, with probability one, tY 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Yc , Yc ](t) 1 2 0 + [Y 1 (s)Y 2 (s) − Y 1 (s−)Y 2 (s−) 0≤s≤t − (Y 1 (s) − Y 1 (s−))Y 2 (s−) − (Y 2 (s) − Y 2 (s−))Y 1 (s−)],from which the required result easily follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 41 / 58
  • 181. Proof. We consider Y 1 and Y 2 as components of a vectorY = (Y 1 , Y 2 ) and we take f in Theorem 7 to be the smooth mappingfrom R2 to R given by f (x 1 , x 2 ) = x 1 x 2 .By Theorem 7, we then obtain, for each t ≥ 0, with probability one, tY 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Yc , Yc ](t) 1 2 0 + [Y 1 (s)Y 2 (s) − Y 1 (s−)Y 2 (s−) 0≤s≤t − (Y 1 (s) − Y 1 (s−))Y 2 (s−) − (Y 2 (s) − Y 2 (s−))Y 1 (s−)],from which the required result easily follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 41 / 58
  • 182. Proof. We consider Y 1 and Y 2 as components of a vectorY = (Y 1 , Y 2 ) and we take f in Theorem 7 to be the smooth mappingfrom R2 to R given by f (x 1 , x 2 ) = x 1 x 2 .By Theorem 7, we then obtain, for each t ≥ 0, with probability one, tY 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Yc , Yc ](t) 1 2 0 + [Y 1 (s)Y 2 (s) − Y 1 (s−)Y 2 (s−) 0≤s≤t − (Y 1 (s) − Y 1 (s−))Y 2 (s−) − (Y 2 (s) − Y 2 (s−))Y 1 (s−)],from which the required result easily follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 41 / 58
  • 183. Proof. We consider Y 1 and Y 2 as components of a vectorY = (Y 1 , Y 2 ) and we take f in Theorem 7 to be the smooth mappingfrom R2 to R given by f (x 1 , x 2 ) = x 1 x 2 .By Theorem 7, we then obtain, for each t ≥ 0, with probability one, tY 1 (t)Y 2 (t) = Y 1 (0)Y 2 (0) + Y 1 (s−)dY 2 (s) 0 t + Y 2 (s−)dY 1 (s) + [Yc , Yc ](t) 1 2 0 + [Y 1 (s)Y 2 (s) − Y 1 (s−)Y 2 (s−) 0≤s≤t − (Y 1 (s) − Y 1 (s−))Y 2 (s−) − (Y 2 (s) − Y 2 (s−))Y 1 (s−)],from which the required result easily follows. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 41 / 58
  • 184. We can learn much about the way our Itô formulae work by writing theproduct formula in differential form:- d(Y 1 (t)Y 2 (t)) = Y 1 (t−)dY 2 (t) + Y 2 (t−)dY 1 (t) + d[Y 1 , Y 2 ](t).We see that the term d[Y 1 , Y 2 ](t), which is sometimes called an Itôcorrection, arises as a result of the following formal product relationsbetween differentials:- dB i (t)dB j (t) = δ ij dt ; N(dt, dx)N(dt, dy ) = N(dt, dx)δ(x − y ),for 1 ≤ i, j ≤ m, with all other products of differentials vanishing and ifyou have little previous experience of this game, these relations are avery valuable guide to intuition. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 42 / 58
  • 185. We can learn much about the way our Itô formulae work by writing theproduct formula in differential form:- d(Y 1 (t)Y 2 (t)) = Y 1 (t−)dY 2 (t) + Y 2 (t−)dY 1 (t) + d[Y 1 , Y 2 ](t).We see that the term d[Y 1 , Y 2 ](t), which is sometimes called an Itôcorrection, arises as a result of the following formal product relationsbetween differentials:- dB i (t)dB j (t) = δ ij dt ; N(dt, dx)N(dt, dy ) = N(dt, dx)δ(x − y ),for 1 ≤ i, j ≤ m, with all other products of differentials vanishing and ifyou have little previous experience of this game, these relations are avery valuable guide to intuition. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 42 / 58
  • 186. We can learn much about the way our Itô formulae work by writing theproduct formula in differential form:- d(Y 1 (t)Y 2 (t)) = Y 1 (t−)dY 2 (t) + Y 2 (t−)dY 1 (t) + d[Y 1 , Y 2 ](t).We see that the term d[Y 1 , Y 2 ](t), which is sometimes called an Itôcorrection, arises as a result of the following formal product relationsbetween differentials:- dB i (t)dB j (t) = δ ij dt ; N(dt, dx)N(dt, dy ) = N(dt, dx)δ(x − y ),for 1 ≤ i, j ≤ m, with all other products of differentials vanishing and ifyou have little previous experience of this game, these relations are avery valuable guide to intuition. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 42 / 58
  • 187. We can learn much about the way our Itô formulae work by writing theproduct formula in differential form:- d(Y 1 (t)Y 2 (t)) = Y 1 (t−)dY 2 (t) + Y 2 (t−)dY 1 (t) + d[Y 1 , Y 2 ](t).We see that the term d[Y 1 , Y 2 ](t), which is sometimes called an Itôcorrection, arises as a result of the following formal product relationsbetween differentials:- dB i (t)dB j (t) = δ ij dt ; N(dt, dx)N(dt, dy ) = N(dt, dx)δ(x − y ),for 1 ≤ i, j ≤ m, with all other products of differentials vanishing and ifyou have little previous experience of this game, these relations are avery valuable guide to intuition. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 42 / 58
  • 188. We can learn much about the way our Itô formulae work by writing theproduct formula in differential form:- d(Y 1 (t)Y 2 (t)) = Y 1 (t−)dY 2 (t) + Y 2 (t−)dY 1 (t) + d[Y 1 , Y 2 ](t).We see that the term d[Y 1 , Y 2 ](t), which is sometimes called an Itôcorrection, arises as a result of the following formal product relationsbetween differentials:- dB i (t)dB j (t) = δ ij dt ; N(dt, dx)N(dt, dy ) = N(dt, dx)δ(x − y ),for 1 ≤ i, j ≤ m, with all other products of differentials vanishing and ifyou have little previous experience of this game, these relations are avery valuable guide to intuition. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 42 / 58
  • 189. For completeness, we will give another characterisation of quadraticvariation which is sometimes quite useful. We recall the sequence ofpartitions (Pn , n ∈ N), with mesh tending to zero which wereintroduced earlier.TheoremIf X and Y are real-valued Lévy-type stochastic integrals then for eacht ≥ 0, with probability one, we have mn (n) (n) (n) (n) [X , Y ](t) = lim (X (tj+1 ) − X (tj ))(Y (tj+1 ) − Y (tj )), n→∞ j=0where the limit is taken in probability. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 43 / 58
  • 190. For completeness, we will give another characterisation of quadraticvariation which is sometimes quite useful. We recall the sequence ofpartitions (Pn , n ∈ N), with mesh tending to zero which wereintroduced earlier.TheoremIf X and Y are real-valued Lévy-type stochastic integrals then for eacht ≥ 0, with probability one, we have mn (n) (n) (n) (n) [X , Y ](t) = lim (X (tj+1 ) − X (tj ))(Y (tj+1 ) − Y (tj )), n→∞ j=0where the limit is taken in probability. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 43 / 58
  • 191. For completeness, we will give another characterisation of quadraticvariation which is sometimes quite useful. We recall the sequence ofpartitions (Pn , n ∈ N), with mesh tending to zero which wereintroduced earlier.TheoremIf X and Y are real-valued Lévy-type stochastic integrals then for eacht ≥ 0, with probability one, we have mn (n) (n) (n) (n) [X , Y ](t) = lim (X (tj+1 ) − X (tj ))(Y (tj+1 ) − Y (tj )), n→∞ j=0where the limit is taken in probability. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 43 / 58
  • 192. For completeness, we will give another characterisation of quadraticvariation which is sometimes quite useful. We recall the sequence ofpartitions (Pn , n ∈ N), with mesh tending to zero which wereintroduced earlier.TheoremIf X and Y are real-valued Lévy-type stochastic integrals then for eacht ≥ 0, with probability one, we have mn (n) (n) (n) (n) [X , Y ](t) = lim (X (tj+1 ) − X (tj ))(Y (tj+1 ) − Y (tj )), n→∞ j=0where the limit is taken in probability. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 43 / 58
  • 193. Proof. By polarisation, it is sufficient to consider the case X = Y .Using the identity (x − y )2 = x 2 − y 2 − 2y (x − y )for x, y ∈ R, we deduce that mn mn mn (n) (n) (n) (n) (X (tj+1 ) − X (tj ))2 = X (tj+1 )2 − X (tj )2 j=0 j=0 j=0 mn (n) (n) (n) − 2 X (tj )(X (tj+1 ) − X (tj )), j=0and the required result follows from Itô’s product formula. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 44 / 58
  • 194. Proof. By polarisation, it is sufficient to consider the case X = Y .Using the identity (x − y )2 = x 2 − y 2 − 2y (x − y )for x, y ∈ R, we deduce that mn mn mn (n) (n) (n) (n) (X (tj+1 ) − X (tj ))2 = X (tj+1 )2 − X (tj )2 j=0 j=0 j=0 mn (n) (n) (n) − 2 X (tj )(X (tj+1 ) − X (tj )), j=0and the required result follows from Itô’s product formula. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 44 / 58
  • 195. Proof. By polarisation, it is sufficient to consider the case X = Y .Using the identity (x − y )2 = x 2 − y 2 − 2y (x − y )for x, y ∈ R, we deduce that mn mn mn (n) (n) (n) (n) (X (tj+1 ) − X (tj ))2 = X (tj+1 )2 − X (tj )2 j=0 j=0 j=0 mn (n) (n) (n) − 2 X (tj )(X (tj+1 ) − X (tj )), j=0and the required result follows from Itô’s product formula. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 44 / 58
  • 196. Many of the results of this lecture extend from Lévy-type stochasticintegrals to arbitrary semimartingales. In particular, if F is a simpleprocess and X is a semimartingale we can again use Itô’s prescriptionto define t F (s)dX (s) = F (tj )(X (tj+1 ) − X (tj )), 0and then pass to the limit to obtain more general stochastic integrals.Itô’s formula can be established in the form given in Theorem 7 and thequadratic variation of semimartingales defined as the correction termin the corresponding Itô product formula.Although stochastic calculus for general semimartingales is not thesubject of these lectures, we do require one result - the famous Lévycharacterisation of Brownian motion. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 45 / 58
  • 197. Many of the results of this lecture extend from Lévy-type stochasticintegrals to arbitrary semimartingales. In particular, if F is a simpleprocess and X is a semimartingale we can again use Itô’s prescriptionto define t F (s)dX (s) = F (tj )(X (tj+1 ) − X (tj )), 0and then pass to the limit to obtain more general stochastic integrals.Itô’s formula can be established in the form given in Theorem 7 and thequadratic variation of semimartingales defined as the correction termin the corresponding Itô product formula.Although stochastic calculus for general semimartingales is not thesubject of these lectures, we do require one result - the famous Lévycharacterisation of Brownian motion. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 45 / 58
  • 198. Many of the results of this lecture extend from Lévy-type stochasticintegrals to arbitrary semimartingales. In particular, if F is a simpleprocess and X is a semimartingale we can again use Itô’s prescriptionto define t F (s)dX (s) = F (tj )(X (tj+1 ) − X (tj )), 0and then pass to the limit to obtain more general stochastic integrals.Itô’s formula can be established in the form given in Theorem 7 and thequadratic variation of semimartingales defined as the correction termin the corresponding Itô product formula.Although stochastic calculus for general semimartingales is not thesubject of these lectures, we do require one result - the famous Lévycharacterisation of Brownian motion. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 45 / 58
  • 199. Many of the results of this lecture extend from Lévy-type stochasticintegrals to arbitrary semimartingales. In particular, if F is a simpleprocess and X is a semimartingale we can again use Itô’s prescriptionto define t F (s)dX (s) = F (tj )(X (tj+1 ) − X (tj )), 0and then pass to the limit to obtain more general stochastic integrals.Itô’s formula can be established in the form given in Theorem 7 and thequadratic variation of semimartingales defined as the correction termin the corresponding Itô product formula.Although stochastic calculus for general semimartingales is not thesubject of these lectures, we do require one result - the famous Lévycharacterisation of Brownian motion. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 45 / 58
  • 200. Many of the results of this lecture extend from Lévy-type stochasticintegrals to arbitrary semimartingales. In particular, if F is a simpleprocess and X is a semimartingale we can again use Itô’s prescriptionto define t F (s)dX (s) = F (tj )(X (tj+1 ) − X (tj )), 0and then pass to the limit to obtain more general stochastic integrals.Itô’s formula can be established in the form given in Theorem 7 and thequadratic variation of semimartingales defined as the correction termin the corresponding Itô product formula.Although stochastic calculus for general semimartingales is not thesubject of these lectures, we do require one result - the famous Lévycharacterisation of Brownian motion. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 45 / 58
  • 201. Theorem (Lévy’s characterisation)Let M = (M(t), t ≥ 0) be a continuous centered martingale, which isadapted to a given filtration (Ft , t ≥ 0). If [Mi , Mj ](t) = aij t for eacht ≥ 0, 1 ≤ i, j ≤ d where a = (aij ) is a positive definite symmetricmatrix, then M is an Ft -adapted Brownian motion with covariance a.Proof. Fix u ∈ Rd and define the process (Yu (t), t ≥ 0) byYu (t) = ei(u,M(t)) , then by Itô’s formula, we obtain 1 dYu (t) = iu j Yu (t)dMj (t) − u i u j Yu (t)d[Mi , Mj ](t) 2 1 = iu j Yu (t)dMj (t) − (u, au)Yu (t)dt. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 46 / 58
  • 202. Theorem (Lévy’s characterisation)Let M = (M(t), t ≥ 0) be a continuous centered martingale, which isadapted to a given filtration (Ft , t ≥ 0). If [Mi , Mj ](t) = aij t for eacht ≥ 0, 1 ≤ i, j ≤ d where a = (aij ) is a positive definite symmetricmatrix, then M is an Ft -adapted Brownian motion with covariance a.Proof. Fix u ∈ Rd and define the process (Yu (t), t ≥ 0) byYu (t) = ei(u,M(t)) , then by Itô’s formula, we obtain 1 dYu (t) = iu j Yu (t)dMj (t) − u i u j Yu (t)d[Mi , Mj ](t) 2 1 = iu j Yu (t)dMj (t) − (u, au)Yu (t)dt. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 46 / 58
  • 203. Theorem (Lévy’s characterisation)Let M = (M(t), t ≥ 0) be a continuous centered martingale, which isadapted to a given filtration (Ft , t ≥ 0). If [Mi , Mj ](t) = aij t for eacht ≥ 0, 1 ≤ i, j ≤ d where a = (aij ) is a positive definite symmetricmatrix, then M is an Ft -adapted Brownian motion with covariance a.Proof. Fix u ∈ Rd and define the process (Yu (t), t ≥ 0) byYu (t) = ei(u,M(t)) , then by Itô’s formula, we obtain 1 dYu (t) = iu j Yu (t)dMj (t) − u i u j Yu (t)d[Mi , Mj ](t) 2 1 = iu j Yu (t)dMj (t) − (u, au)Yu (t)dt. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 46 / 58
  • 204. Theorem (Lévy’s characterisation)Let M = (M(t), t ≥ 0) be a continuous centered martingale, which isadapted to a given filtration (Ft , t ≥ 0). If [Mi , Mj ](t) = aij t for eacht ≥ 0, 1 ≤ i, j ≤ d where a = (aij ) is a positive definite symmetricmatrix, then M is an Ft -adapted Brownian motion with covariance a.Proof. Fix u ∈ Rd and define the process (Yu (t), t ≥ 0) byYu (t) = ei(u,M(t)) , then by Itô’s formula, we obtain 1 dYu (t) = iu j Yu (t)dMj (t) − u i u j Yu (t)d[Mi , Mj ](t) 2 1 = iu j Yu (t)dMj (t) − (u, au)Yu (t)dt. 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 46 / 58
  • 205. Upon integrating from s to t, we obtain t t 1 Yu (t) = Yu (s) + iu j Yu (τ )dMj (τ ) − (u, au) Yu (τ )dτ. s 2 sNow take conditional expectations of both sides with respect to Fs , anduse the conditional Fubini Theorem to obtain t 1 E(Yu (t)|Fs ) = Yu (s) − (u, au) E(Yu (τ )|Fs )dτ. 2 s 1 Hence E(ei(u,M(t)−M(s)) |Fs ) = e− 2 (u,au)(t−s) . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 47 / 58
  • 206. Upon integrating from s to t, we obtain t t 1 Yu (t) = Yu (s) + iu j Yu (τ )dMj (τ ) − (u, au) Yu (τ )dτ. s 2 sNow take conditional expectations of both sides with respect to Fs , anduse the conditional Fubini Theorem to obtain t 1 E(Yu (t)|Fs ) = Yu (s) − (u, au) E(Yu (τ )|Fs )dτ. 2 s 1 Hence E(ei(u,M(t)−M(s)) |Fs ) = e− 2 (u,au)(t−s) . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 47 / 58
  • 207. Upon integrating from s to t, we obtain t t 1 Yu (t) = Yu (s) + iu j Yu (τ )dMj (τ ) − (u, au) Yu (τ )dτ. s 2 sNow take conditional expectations of both sides with respect to Fs , anduse the conditional Fubini Theorem to obtain t 1 E(Yu (t)|Fs ) = Yu (s) − (u, au) E(Yu (τ )|Fs )dτ. 2 s 1 Hence E(ei(u,M(t)−M(s)) |Fs ) = e− 2 (u,au)(t−s) . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 47 / 58
  • 208. Stochastic Differential EquationsUsing Picard iteration one can show the existence of a unique solutiontodY (t) = b(Y (t−))dt + σ(Y (t−))dB(t) + (0.10) + ˜ F (Y (t−), x)N(dt, dx) + G(Y (t−), x)N(dt, dx), |x|<c |x|≥cwhich is a convenient shorthand for the system of SDE’s:-dY i (t) = bi (Y (t−))dt + σji (Y (t−))dB j (t) + (0.11) + ˜ F i (Y (t−), x)N(dt, dx) + Gi (Y (t−), x)N(dt, dx), |x|≤c |x|>cwhere each 1 ≤ i ≤ d. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 48 / 58
  • 209. Stochastic Differential EquationsUsing Picard iteration one can show the existence of a unique solutiontodY (t) = b(Y (t−))dt + σ(Y (t−))dB(t) + (0.10) + ˜ F (Y (t−), x)N(dt, dx) + G(Y (t−), x)N(dt, dx), |x|<c |x|≥cwhich is a convenient shorthand for the system of SDE’s:-dY i (t) = bi (Y (t−))dt + σji (Y (t−))dB j (t) + (0.11) + ˜ F i (Y (t−), x)N(dt, dx) + Gi (Y (t−), x)N(dt, dx), |x|≤c |x|>cwhere each 1 ≤ i ≤ d. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 48 / 58
  • 210. Stochastic Differential EquationsUsing Picard iteration one can show the existence of a unique solutiontodY (t) = b(Y (t−))dt + σ(Y (t−))dB(t) + (0.10) + ˜ F (Y (t−), x)N(dt, dx) + G(Y (t−), x)N(dt, dx), |x|<c |x|≥cwhich is a convenient shorthand for the system of SDE’s:-dY i (t) = bi (Y (t−))dt + σji (Y (t−))dB j (t) + (0.11) + ˜ F i (Y (t−), x)N(dt, dx) + Gi (Y (t−), x)N(dt, dx), |x|≤c |x|>cwhere each 1 ≤ i ≤ d. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 48 / 58
  • 211. Stochastic Differential EquationsUsing Picard iteration one can show the existence of a unique solutiontodY (t) = b(Y (t−))dt + σ(Y (t−))dB(t) + (0.10) + ˜ F (Y (t−), x)N(dt, dx) + G(Y (t−), x)N(dt, dx), |x|<c |x|≥cwhich is a convenient shorthand for the system of SDE’s:-dY i (t) = bi (Y (t−))dt + σji (Y (t−))dB j (t) + (0.11) + ˜ F i (Y (t−), x)N(dt, dx) + Gi (Y (t−), x)N(dt, dx), |x|≤c |x|>cwhere each 1 ≤ i ≤ d. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 48 / 58
  • 212. The simplest conditions under which this holds are:-(1) Lipschitz ConditionThere exists K1 > 0 such that for all y1 , y2 ∈ Rd , |b(y1 ) − b(y2 )|2 + ||a(y1 , y1 ) − 2a(y1 , y2 ) + a(y2 , y2 )||(0.12) + |F (y1 , x) − F (y2 , x)|2 ν(dx) ≤ K1 |y1 − y2 |2 . |x|<c(2) Growth ConditionThere exists K2 > 0 such that for all y ∈ Rd , |b(y )|2 + ||a(y , y )|| + |F (y , x)|2 ν(dx) ≤ K2 (1 + |y |2 ). (0.13) |x|<c(3) Big Jumps ConditionG is jointly measurable and y → G(y , x) is continuous for all |x| ≥ 1. d iHere, || · || is the matrix seminorm ||a|| = i=1 |ai |, anda(x, y ) = σ(x)σ(y )T . Dave Applebaum (Sheffield UK) Lecture 4 December 2011 49 / 58
  • 213. The simplest conditions under which this holds are:-(1) Lipschitz ConditionThere exists K1 > 0 such that for all y1 , y2 ∈ Rd , |b(y1 ) − b(y2 )|2 + ||a(y1 , y1 ) − 2a(y1 , y2 ) + a(y2 , y2 )||(0.12) + |F (y1 , x) − F (y2 , x)|2 ν(dx) ≤ K1 |y1 − y2 |2 . |x|<c(2) Growth ConditionThere exists K2 > 0 such that for all y ∈ Rd , |b(y )|2 + ||a(y , y )|| + |F (y , x)|2 ν(dx) ≤ K2 (1 + |y |2 ). (0.13) |x|<c(3) Big Jumps ConditionG is jointly measurable and y → G(y , x) is continuous for all |x| ≥ 1. d iHere, || · || is the matrix seminorm ||a|| = i=1 |ai |, anda(x, y ) = σ(x)σ(y )T . Dave Applebaum (Sheffield UK) Lecture 4 December 2011 49 / 58
  • 214. The simplest conditions under which this holds are:-(1) Lipschitz ConditionThere exists K1 > 0 such that for all y1 , y2 ∈ Rd , |b(y1 ) − b(y2 )|2 + ||a(y1 , y1 ) − 2a(y1 , y2 ) + a(y2 , y2 )||(0.12) + |F (y1 , x) − F (y2 , x)|2 ν(dx) ≤ K1 |y1 − y2 |2 . |x|<c(2) Growth ConditionThere exists K2 > 0 such that for all y ∈ Rd , |b(y )|2 + ||a(y , y )|| + |F (y , x)|2 ν(dx) ≤ K2 (1 + |y |2 ). (0.13) |x|<c(3) Big Jumps ConditionG is jointly measurable and y → G(y , x) is continuous for all |x| ≥ 1. d iHere, || · || is the matrix seminorm ||a|| = i=1 |ai |, anda(x, y ) = σ(x)σ(y )T . Dave Applebaum (Sheffield UK) Lecture 4 December 2011 49 / 58
  • 215. We also impose the standard initial condition Y (0) = Y0 (a.s.) forwhich Y0 is independent of (Ft , t > 0). Solutions of SDEs are Markovprocesses and, in the case where there are no jumps, diffusionprocesses.A special case of considerable interest is dY (t) = L(Y (t−))dX (t).You can check that the conditions given above boil down to the singlerequirement that L be globally Lipshitz, in order to get existence anduniqueness. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 50 / 58
  • 216. We also impose the standard initial condition Y (0) = Y0 (a.s.) forwhich Y0 is independent of (Ft , t > 0). Solutions of SDEs are Markovprocesses and, in the case where there are no jumps, diffusionprocesses.A special case of considerable interest is dY (t) = L(Y (t−))dX (t).You can check that the conditions given above boil down to the singlerequirement that L be globally Lipshitz, in order to get existence anduniqueness. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 50 / 58
  • 217. We also impose the standard initial condition Y (0) = Y0 (a.s.) forwhich Y0 is independent of (Ft , t > 0). Solutions of SDEs are Markovprocesses and, in the case where there are no jumps, diffusionprocesses.A special case of considerable interest is dY (t) = L(Y (t−))dX (t).You can check that the conditions given above boil down to the singlerequirement that L be globally Lipshitz, in order to get existence anduniqueness. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 50 / 58
  • 218. Stochastic ExponentialsFor convenience we take d = 1 and consider the problem of finding anadapted process Z = (Z (t), t ≥ 0) which has a stochastic differential dZ (t) = Z (t−)dY (t),where Y is a Lévy-type stochastic integral.The solution of this problem is obtained as follows. We take Z to bethe stochastic exponential (sometimes called Doléans-Dadeexponential after its discoverer), which is denoted asEY = (EY (t), t ≥ 0) and defined as 1 EY (t) = exp Y (t) − [Yc , Yc ](t) (1 + ∆Y (s))e−∆Y (s) , (0.14) 2 0≤s≤tfor each t ≥ 0.We will need the following assumption: (SE) inf{∆Y (t), t > 0} > −1, (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 51 / 58
  • 219. Stochastic ExponentialsFor convenience we take d = 1 and consider the problem of finding anadapted process Z = (Z (t), t ≥ 0) which has a stochastic differential dZ (t) = Z (t−)dY (t),where Y is a Lévy-type stochastic integral.The solution of this problem is obtained as follows. We take Z to bethe stochastic exponential (sometimes called Doléans-Dadeexponential after its discoverer), which is denoted asEY = (EY (t), t ≥ 0) and defined as 1 EY (t) = exp Y (t) − [Yc , Yc ](t) (1 + ∆Y (s))e−∆Y (s) , (0.14) 2 0≤s≤tfor each t ≥ 0.We will need the following assumption: (SE) inf{∆Y (t), t > 0} > −1, (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 51 / 58
  • 220. Stochastic ExponentialsFor convenience we take d = 1 and consider the problem of finding anadapted process Z = (Z (t), t ≥ 0) which has a stochastic differential dZ (t) = Z (t−)dY (t),where Y is a Lévy-type stochastic integral.The solution of this problem is obtained as follows. We take Z to bethe stochastic exponential (sometimes called Doléans-Dadeexponential after its discoverer), which is denoted asEY = (EY (t), t ≥ 0) and defined as 1 EY (t) = exp Y (t) − [Yc , Yc ](t) (1 + ∆Y (s))e−∆Y (s) , (0.14) 2 0≤s≤tfor each t ≥ 0.We will need the following assumption: (SE) inf{∆Y (t), t > 0} > −1, (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 51 / 58
  • 221. TheoremIf Y is a Lévy-type stochastic integral of the form (0.7) and (SE) holds,then each EY (t) is almost surely finite.Proof. We must show that the infinite product in (0.14) convergesalmost surely. We write (1 + ∆Y (s))e−∆Y (s) = A(t) + B(t), 0≤s≤twhere A(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|≥ 1 } and 2B(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|< 1 } . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 52 / 58
  • 222. TheoremIf Y is a Lévy-type stochastic integral of the form (0.7) and (SE) holds,then each EY (t) is almost surely finite.Proof. We must show that the infinite product in (0.14) convergesalmost surely. We write (1 + ∆Y (s))e−∆Y (s) = A(t) + B(t), 0≤s≤twhere A(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|≥ 1 } and 2B(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|< 1 } . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 52 / 58
  • 223. TheoremIf Y is a Lévy-type stochastic integral of the form (0.7) and (SE) holds,then each EY (t) is almost surely finite.Proof. We must show that the infinite product in (0.14) convergesalmost surely. We write (1 + ∆Y (s))e−∆Y (s) = A(t) + B(t), 0≤s≤twhere A(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|≥ 1 } and 2B(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|< 1 } . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 52 / 58
  • 224. TheoremIf Y is a Lévy-type stochastic integral of the form (0.7) and (SE) holds,then each EY (t) is almost surely finite.Proof. We must show that the infinite product in (0.14) convergesalmost surely. We write (1 + ∆Y (s))e−∆Y (s) = A(t) + B(t), 0≤s≤twhere A(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|≥ 1 } and 2B(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|< 1 } . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 52 / 58
  • 225. TheoremIf Y is a Lévy-type stochastic integral of the form (0.7) and (SE) holds,then each EY (t) is almost surely finite.Proof. We must show that the infinite product in (0.14) convergesalmost surely. We write (1 + ∆Y (s))e−∆Y (s) = A(t) + B(t), 0≤s≤twhere A(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|≥ 1 } and 2B(t) = 0≤s≤t (1 + ∆Y (s))e−∆Y (s) 1{|∆Y (s)|< 1 } . 2 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 52 / 58
  • 226. Now since Y is càdlàg, #{0 ≤ s ≤ t; |∆Y (s)| ≥ 1 } < ∞ (a.s.), and so 2A(t) is a finite product. Using the assumption (SE), we have     B(t) = exp [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } .  2  0≤s≤tWe now employ Taylor’s theorem to obtain the inequality log(1 + y ) − y ≤ Ky 2 ,where K > 0, which is valid whenever |y | < 1 . Hence 2 [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } ≤ |∆Y (s)|2 1{|∆Y (s)| 20≤s≤t 0≤s≤t < ∞ a.s.and we have our required result. 2Of course (SE) ensures that EY (t) > 0 (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 53 / 58
  • 227. Now since Y is càdlàg, #{0 ≤ s ≤ t; |∆Y (s)| ≥ 1 } < ∞ (a.s.), and so 2A(t) is a finite product. Using the assumption (SE), we have     B(t) = exp [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } .  2  0≤s≤tWe now employ Taylor’s theorem to obtain the inequality log(1 + y ) − y ≤ Ky 2 ,where K > 0, which is valid whenever |y | < 1 . Hence 2 [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } ≤ |∆Y (s)|2 1{|∆Y (s)| 20≤s≤t 0≤s≤t < ∞ a.s.and we have our required result. 2Of course (SE) ensures that EY (t) > 0 (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 53 / 58
  • 228. Now since Y is càdlàg, #{0 ≤ s ≤ t; |∆Y (s)| ≥ 1 } < ∞ (a.s.), and so 2A(t) is a finite product. Using the assumption (SE), we have     B(t) = exp [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } .  2  0≤s≤tWe now employ Taylor’s theorem to obtain the inequality log(1 + y ) − y ≤ Ky 2 ,where K > 0, which is valid whenever |y | < 1 . Hence 2 [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } ≤ |∆Y (s)|2 1{|∆Y (s)| 20≤s≤t 0≤s≤t < ∞ a.s.and we have our required result. 2Of course (SE) ensures that EY (t) > 0 (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 53 / 58
  • 229. Now since Y is càdlàg, #{0 ≤ s ≤ t; |∆Y (s)| ≥ 1 } < ∞ (a.s.), and so 2A(t) is a finite product. Using the assumption (SE), we have     B(t) = exp [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } .  2  0≤s≤tWe now employ Taylor’s theorem to obtain the inequality log(1 + y ) − y ≤ Ky 2 ,where K > 0, which is valid whenever |y | < 1 . Hence 2 [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } ≤ |∆Y (s)|2 1{|∆Y (s)| 20≤s≤t 0≤s≤t < ∞ a.s.and we have our required result. 2Of course (SE) ensures that EY (t) > 0 (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 53 / 58
  • 230. Now since Y is càdlàg, #{0 ≤ s ≤ t; |∆Y (s)| ≥ 1 } < ∞ (a.s.), and so 2A(t) is a finite product. Using the assumption (SE), we have     B(t) = exp [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } .  2  0≤s≤tWe now employ Taylor’s theorem to obtain the inequality log(1 + y ) − y ≤ Ky 2 ,where K > 0, which is valid whenever |y | < 1 . Hence 2 [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } ≤ |∆Y (s)|2 1{|∆Y (s)| 20≤s≤t 0≤s≤t < ∞ a.s.and we have our required result. 2Of course (SE) ensures that EY (t) > 0 (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 53 / 58
  • 231. Now since Y is càdlàg, #{0 ≤ s ≤ t; |∆Y (s)| ≥ 1 } < ∞ (a.s.), and so 2A(t) is a finite product. Using the assumption (SE), we have     B(t) = exp [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } .  2  0≤s≤tWe now employ Taylor’s theorem to obtain the inequality log(1 + y ) − y ≤ Ky 2 ,where K > 0, which is valid whenever |y | < 1 . Hence 2 [log(1 + ∆Y (s)) − ∆Y (s)]1{|∆Y (s)|< 1 } ≤ |∆Y (s)|2 1{|∆Y (s)| 20≤s≤t 0≤s≤t < ∞ a.s.and we have our required result. 2Of course (SE) ensures that EY (t) > 0 (a.s.). Dave Applebaum (Sheffield UK) Lecture 4 December 2011 53 / 58
  • 232. The stochastic exponential is, in fact the unique solution of thestochastic differential equation dZ (t) = Z (t−)dY (t), with initialcondition Z (0) = 1 (a.s.).The restrictions (SE) can be dropped and the stochastic exponentialextended to the case where Y is an arbitrary (real valued or evencomplex-valued) càdlàg semimartingale, but the price we have to payis that EY may then take negative values. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 54 / 58
  • 233. The stochastic exponential is, in fact the unique solution of thestochastic differential equation dZ (t) = Z (t−)dY (t), with initialcondition Z (0) = 1 (a.s.).The restrictions (SE) can be dropped and the stochastic exponentialextended to the case where Y is an arbitrary (real valued or evencomplex-valued) càdlàg semimartingale, but the price we have to payis that EY may then take negative values. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 54 / 58
  • 234. The following alternative form of (0.14) is quite useful : EY (t) = eSY (t) , 1 where dSY (t) = F (t)dB(t) + G(t) − F (t)2 dt 2 + log(1 + K (t, x))N(dt, dx) |x|≥1 + ˜ log(1 + H(t, x))N(dt, dx) |x|<1 + (log(1 + H(t, x)) − H(t, x))ν(dx)ds. |x|<1 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 55 / 58
  • 235. The following alternative form of (0.14) is quite useful : EY (t) = eSY (t) , 1 where dSY (t) = F (t)dB(t) + G(t) − F (t)2 dt 2 + log(1 + K (t, x))N(dt, dx) |x|≥1 + ˜ log(1 + H(t, x))N(dt, dx) |x|<1 + (log(1 + H(t, x)) − H(t, x))ν(dx)ds. |x|<1 Dave Applebaum (Sheffield UK) Lecture 4 December 2011 55 / 58
  • 236. Theorem dEY (t) = EY (t−)dY (t).Proof. We apply Itô’s formula to EY (t) = eSY (t) to obtain for eacht ≥ 0, dEY (t) = EY (t−) (F (t)dB(t) + G(t)dt + (log(1 + H(t, x)) − H(t, x))ν(dx)dt) |x|<1 + [exp{SY (t−) + log(1 + K (t, x))} − exp (SY (t−))]N(dt, dx) |x|≥1 + ˜ [exp{SY (t−) + log(1 + H(t, x))} − exp (SY (t−))]N(dt, dx) |x|<1 + [exp{SY (t−) + log(1 + H(t, x))} − exp (SY (t−)) |x|<1 − log(1 + H(t, x)) exp SY (t−)]ν(dx)dt) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 56 / 58
  • 237. Theorem dEY (t) = EY (t−)dY (t).Proof. We apply Itô’s formula to EY (t) = eSY (t) to obtain for eacht ≥ 0, dEY (t) = EY (t−) (F (t)dB(t) + G(t)dt + (log(1 + H(t, x)) − H(t, x))ν(dx)dt) |x|<1 + [exp{SY (t−) + log(1 + K (t, x))} − exp (SY (t−))]N(dt, dx) |x|≥1 + ˜ [exp{SY (t−) + log(1 + H(t, x))} − exp (SY (t−))]N(dt, dx) |x|<1 + [exp{SY (t−) + log(1 + H(t, x))} − exp (SY (t−)) |x|<1 − log(1 + H(t, x)) exp SY (t−)]ν(dx)dt) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 56 / 58
  • 238. Theorem dEY (t) = EY (t−)dY (t).Proof. We apply Itô’s formula to EY (t) = eSY (t) to obtain for eacht ≥ 0, dEY (t) = EY (t−) (F (t)dB(t) + G(t)dt + (log(1 + H(t, x)) − H(t, x))ν(dx)dt) |x|<1 + [exp{SY (t−) + log(1 + K (t, x))} − exp (SY (t−))]N(dt, dx) |x|≥1 + ˜ [exp{SY (t−) + log(1 + H(t, x))} − exp (SY (t−))]N(dt, dx) |x|<1 + [exp{SY (t−) + log(1 + H(t, x))} − exp (SY (t−)) |x|<1 − log(1 + H(t, x)) exp SY (t−)]ν(dx)dt) Dave Applebaum (Sheffield UK) Lecture 4 December 2011 56 / 58
  • 239. and so ˜dEY (t) = EY (t−)[F (t)dB(t)+G(t)dt+K (t, x)N(dt, dx)+H(t, x)N(dt, dx)].as required. 2Examples 1 If Y (t) = σB(t) where σ > 0 and B = (B(t), t ≥ 0) is a standard Brownian motion, then 1 EY (t) = exp σB(t) − σ 2 t . 2 2 If Y = (Y (t), t ≥ 0) is a compound Poisson process, so that each Y (t) = X1 + · · · + XN(t) , where (Xn , n ∈ N) are i.i.d. and N is an independent Poisson process, we have N(t) EY (t) = (1 + Xj ), j=1 for each t ≥ 0. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 57 / 58
  • 240. and so ˜dEY (t) = EY (t−)[F (t)dB(t)+G(t)dt+K (t, x)N(dt, dx)+H(t, x)N(dt, dx)].as required. 2Examples 1 If Y (t) = σB(t) where σ > 0 and B = (B(t), t ≥ 0) is a standard Brownian motion, then 1 EY (t) = exp σB(t) − σ 2 t . 2 2 If Y = (Y (t), t ≥ 0) is a compound Poisson process, so that each Y (t) = X1 + · · · + XN(t) , where (Xn , n ∈ N) are i.i.d. and N is an independent Poisson process, we have N(t) EY (t) = (1 + Xj ), j=1 for each t ≥ 0. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 57 / 58
  • 241. and so ˜dEY (t) = EY (t−)[F (t)dB(t)+G(t)dt+K (t, x)N(dt, dx)+H(t, x)N(dt, dx)].as required. 2Examples 1 If Y (t) = σB(t) where σ > 0 and B = (B(t), t ≥ 0) is a standard Brownian motion, then 1 EY (t) = exp σB(t) − σ 2 t . 2 2 If Y = (Y (t), t ≥ 0) is a compound Poisson process, so that each Y (t) = X1 + · · · + XN(t) , where (Xn , n ∈ N) are i.i.d. and N is an independent Poisson process, we have N(t) EY (t) = (1 + Xj ), j=1 for each t ≥ 0. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 57 / 58
  • 242. and so ˜dEY (t) = EY (t−)[F (t)dB(t)+G(t)dt+K (t, x)N(dt, dx)+H(t, x)N(dt, dx)].as required. 2Examples 1 If Y (t) = σB(t) where σ > 0 and B = (B(t), t ≥ 0) is a standard Brownian motion, then 1 EY (t) = exp σB(t) − σ 2 t . 2 2 If Y = (Y (t), t ≥ 0) is a compound Poisson process, so that each Y (t) = X1 + · · · + XN(t) , where (Xn , n ∈ N) are i.i.d. and N is an independent Poisson process, we have N(t) EY (t) = (1 + Xj ), j=1 for each t ≥ 0. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 57 / 58
  • 243. and so ˜dEY (t) = EY (t−)[F (t)dB(t)+G(t)dt+K (t, x)N(dt, dx)+H(t, x)N(dt, dx)].as required. 2Examples 1 If Y (t) = σB(t) where σ > 0 and B = (B(t), t ≥ 0) is a standard Brownian motion, then 1 EY (t) = exp σB(t) − σ 2 t . 2 2 If Y = (Y (t), t ≥ 0) is a compound Poisson process, so that each Y (t) = X1 + · · · + XN(t) , where (Xn , n ∈ N) are i.i.d. and N is an independent Poisson process, we have N(t) EY (t) = (1 + Xj ), j=1 for each t ≥ 0. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 57 / 58
  • 244. If X and Y are Lévy-type stochastic integrals, you can check that EX (t)EY (t) = EX +Y +[X ,Y ] (t),for each t ≥ 0. Dave Applebaum (Sheffield UK) Lecture 4 December 2011 58 / 58

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