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  • 1. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Large Deviations in Dynamical Systems Part 1: A market model with Markovian switching Terry Lynch1 (joint work with J. Appleby1 , X. Mao2 and H. Wu1 ) 1 Dublin City University, Ireland. 2 Strathclyde University, Glasgow, U.K. Stochastic Evolutionary Problems Theory, Modelling and Numerics University of Chester Oct 31st - Nov 2nd 2007 Supported by the Irish Research Council for Science, Engineering and Technology
  • 2. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline Introduction 1 Mathematical Results 2 Application to Financial Market Models 3 Extensions and future work 4
  • 3. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline Introduction 1 Mathematical Results 2 Application to Financial Market Models 3 Extensions and future work 4
  • 4. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ =c a.s. 2t log log t t→∞ The results are applied to financial market models which are subject to random regime shifts (bearish to bullish) and to changes in market sentiment. We show that the security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  • 5. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ =c a.s. 2t log log t t→∞ The results are applied to financial market models which are subject to random regime shifts (bearish to bullish) and to changes in market sentiment. We show that the security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  • 6. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Introduction We consider the size of the large fluctuations of a stochastic differential equation with Markovian switching, concentrating on processes which obey the law of the iterated logarithm: |X (t)| lim sup √ =c a.s. 2t log log t t→∞ The results are applied to financial market models which are subject to random regime shifts (bearish to bullish) and to changes in market sentiment. We show that the security model exhibits the same long-run growth and deviation properties as conventional geometric Brownian motion.
  • 7. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Classical geometric Brownian motion Characterised as the unique solution of the linear S.D.E dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t) where µ is the instantaneous mean rate of growth of the price, σ its instantaneous volatility and S∗ (0) > 0. S∗ grows exponentially according to log S∗ (t) 1 = µ − σ2, lim a.s. (1) t 2 t→∞ The maximum size of the large deviations from this growth trend obey the law of the iterated logarithm | log S∗ (t) − (µ − 1 σ 2 )t| 2 √ lim sup = σ, a.s. (2) 2t log log t t→∞
  • 8. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Classical geometric Brownian motion Characterised as the unique solution of the linear S.D.E dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t) where µ is the instantaneous mean rate of growth of the price, σ its instantaneous volatility and S∗ (0) > 0. S∗ grows exponentially according to log S∗ (t) 1 = µ − σ2, lim a.s. (1) t 2 t→∞ The maximum size of the large deviations from this growth trend obey the law of the iterated logarithm | log S∗ (t) − (µ − 1 σ 2 )t| 2 √ lim sup = σ, a.s. (2) 2t log log t t→∞
  • 9. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Classical geometric Brownian motion Characterised as the unique solution of the linear S.D.E dS∗ (t) = µS∗ (t) dt + σS∗ (t) dB(t) where µ is the instantaneous mean rate of growth of the price, σ its instantaneous volatility and S∗ (0) > 0. S∗ grows exponentially according to log S∗ (t) 1 = µ − σ2, lim a.s. (1) t 2 t→∞ The maximum size of the large deviations from this growth trend obey the law of the iterated logarithm | log S∗ (t) − (µ − 1 σ 2 )t| 2 √ lim sup = σ, a.s. (2) 2t log log t t→∞
  • 10. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline Introduction 1 Mathematical Results 2 Application to Financial Market Models 3 Extensions and future work 4
  • 11. Introduction Mathematical Results Application to Financial Market Models Extensions and future work S.D.E with Markovian switching We study the stochastic differential equation with Markovian switching dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3) where X (0) = x0 , f , g : R × S × [0, ∞) → R are continuous functions obeying local Lipschitz continuity and linear growth conditions, Y is an irreducible continuous-time Markov chain with finite state space S and is independent of the Brownian motion B. Under these conditions, there exists a unique continuous and adapted process which satisfies (3).
  • 12. Introduction Mathematical Results Application to Financial Market Models Extensions and future work S.D.E with Markovian switching We study the stochastic differential equation with Markovian switching dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3) where X (0) = x0 , f , g : R × S × [0, ∞) → R are continuous functions obeying local Lipschitz continuity and linear growth conditions, Y is an irreducible continuous-time Markov chain with finite state space S and is independent of the Brownian motion B. Under these conditions, there exists a unique continuous and adapted process which satisfies (3).
  • 13. Introduction Mathematical Results Application to Financial Market Models Extensions and future work S.D.E with Markovian switching We study the stochastic differential equation with Markovian switching dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t) (3) where X (0) = x0 , f , g : R × S × [0, ∞) → R are continuous functions obeying local Lipschitz continuity and linear growth conditions, Y is an irreducible continuous-time Markov chain with finite state space S and is independent of the Brownian motion B. Under these conditions, there exists a unique continuous and adapted process which satisfies (3).
  • 14. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 Let X be the unique adapted continuous solution satisfying dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, y , t) ∈ R × S × [0, ∞) 0 < K2 ≤ g 2 (x, y , t) ≤ K1 xf (x, y , t) ≤ ρ, then X satisfies |X (t)| lim sup √ ≤ K1 , a.s. 2t log log t t→∞
  • 15. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 Let X be the unique adapted continuous solution satisfying dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, y , t) ∈ R × S × [0, ∞) 0 < K2 ≤ g 2 (x, y , t) ≤ K1 xf (x, y , t) ≤ ρ, then X satisfies |X (t)| lim sup √ ≤ K1 , a.s. 2t log log t t→∞
  • 16. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 Let X be the unique adapted continuous solution satisfying dX (t) = f (X (t), Y (t), t) dt + g (X (t), Y (t), t) dB(t). If there exist ρ > 0 and real numbers K1 and K2 such that ∀ (x, y , t) ∈ R × S × [0, ∞) 0 < K2 ≤ g 2 (x, y , t) ≤ K1 xf (x, y , t) ≤ ρ, then X satisfies |X (t)| lim sup √ ≤ K1 , a.s. 2t log log t t→∞
  • 17. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 continued If, moreover, there exists an L ∈ R such that xf (x, y , t) 1 =: L > − inf 2 (x, y , t) (x,y ,t)∈R×S×[0,∞) g 2 Then X satisfies |X (t)| lim sup √ ≥ K2 , a.s. 2t log log t t→∞
  • 18. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Main results Theorem 1 continued If, moreover, there exists an L ∈ R such that xf (x, y , t) 1 =: L > − inf 2 (x, y , t) (x,y ,t)∈R×S×[0,∞) g 2 Then X satisfies |X (t)| lim sup √ ≥ K2 , a.s. 2t log log t t→∞
  • 19. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are recurrent and stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  • 20. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are recurrent and stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  • 21. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are recurrent and stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  • 22. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline of proof The proofs rely on time-change and stochastic comparison arguments, constructing upper and lower bounds on |X | which, under an appropriate change of time and scale, are recurrent and stationary processes whose dynamics are not determined by Y . The large deviations of these processes are determined by means of a classical theorem of Motoo.
  • 23. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline Introduction 1 Mathematical Results 2 Application to Financial Market Models 3 Extensions and future work 4
  • 24. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Security Price Model The large deviation results are now applied to a security price model, where the security price S obeys t≥0 dS(t) = µS(t)dt + S(t)dX (t), Here we consider the special case t≥0 dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), with X (0) = 0 and γ : S → R {0}. Under the conditions xf (x, y , t) ≤ ρ and sup γ 2 (y ) (x,y ,t)∈R×S×[0,∞) xf (x, y , t) 1 >− , inf 2 (y ) γ 2 (x,y ,t)∈R×S×[0,∞)
  • 25. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Security Price Model The large deviation results are now applied to a security price model, where the security price S obeys t≥0 dS(t) = µS(t)dt + S(t)dX (t), Here we consider the special case t≥0 dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), with X (0) = 0 and γ : S → R {0}. Under the conditions xf (x, y , t) ≤ ρ and sup γ 2 (y ) (x,y ,t)∈R×S×[0,∞) xf (x, y , t) 1 >− , inf 2 (y ) γ 2 (x,y ,t)∈R×S×[0,∞)
  • 26. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Security Price Model The large deviation results are now applied to a security price model, where the security price S obeys t≥0 dS(t) = µS(t)dt + S(t)dX (t), Here we consider the special case t≥0 dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t), with X (0) = 0 and γ : S → R {0}. Under the conditions xf (x, y , t) ≤ ρ and sup γ 2 (y ) (x,y ,t)∈R×S×[0,∞) xf (x, y , t) 1 >− , inf 2 (y ) γ 2 (x,y ,t)∈R×S×[0,∞)
  • 27. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Fluctuations in the Markovian switching model Theorem 2 Let S be the unique continuous adapted process governed by t≥0 dS(t) = µS(t)dt + S(t)dX (t), with S(0) = s0 > 0, where X is defined on the previous slide. Then: 1 log S(t) 12 = µ − σ∗ , lim a.s. t 2 t→∞ 2 t | log S(t) − (µt − 1 0 γ 2 (Y (s))ds)| 2 √ lim sup = σ∗ 2t log log t t→∞ where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability 2 distribution of Y .
  • 28. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Fluctuations in the Markovian switching model Theorem 2 Let S be the unique continuous adapted process governed by t≥0 dS(t) = µS(t)dt + S(t)dX (t), with S(0) = s0 > 0, where X is defined on the previous slide. Then: 1 log S(t) 12 = µ − σ∗ , lim a.s. t 2 t→∞ 2 t | log S(t) − (µt − 1 0 γ 2 (Y (s))ds)| 2 √ lim sup = σ∗ 2t log log t t→∞ where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability 2 distribution of Y .
  • 29. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Fluctuations in the Markovian switching model Theorem 2 Let S be the unique continuous adapted process governed by t≥0 dS(t) = µS(t)dt + S(t)dX (t), with S(0) = s0 > 0, where X is defined on the previous slide. Then: 1 log S(t) 12 = µ − σ∗ , lim a.s. t 2 t→∞ 2 t | log S(t) − (µt − 1 0 γ 2 (Y (s))ds)| 2 √ lim sup = σ∗ 2t log log t t→∞ where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability 2 distribution of Y .
  • 30. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Comments Despite the presence of the Markov process Y (which introduces regime shifts) and the X -dependent drift term f (which introduces inefficiency), we have shown that the new market model obeys the same asymptotic properties of standard geometric Brownian motion models. However, the analysis is now more complicated because the increments are neither stationary nor Gaussian, and it is not possible to obtain an explicit solution.
  • 31. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Comments Despite the presence of the Markov process Y (which introduces regime shifts) and the X -dependent drift term f (which introduces inefficiency), we have shown that the new market model obeys the same asymptotic properties of standard geometric Brownian motion models. However, the analysis is now more complicated because the increments are neither stationary nor Gaussian, and it is not possible to obtain an explicit solution.
  • 32. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Outline Introduction 1 Mathematical Results 2 Application to Financial Market Models 3 Extensions and future work 4
  • 33. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions and future work We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded. We hope in the future to extend the model to the case of unbounded noise.
  • 34. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions and future work We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded. We hope in the future to extend the model to the case of unbounded noise.
  • 35. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions and future work We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded. We hope in the future to extend the model to the case of unbounded noise.
  • 36. Introduction Mathematical Results Application to Financial Market Models Extensions and future work Extensions and future work We can also get upper and lower bounds on both positive and negative large fluctuations under the assumption that the drift is integrable. A result can be proven about the large fluctuations of the incremental returns when the diffusion coefficient depends on X , Y and t. Can be extended to the case in which the diffusion coefficient in X depends not only on the Markovian switching term but also on a delay term, once that diffusion coefficient remains bounded. We hope in the future to extend the model to the case of unbounded noise.