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Introduction to Stochastic calculus

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Talk given at Morgan Stanley and at the IITs to introduce students to stochastic calculus

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Introduction to Stochastic calculus

1. 1. Introduction to Stochastic Calculus Introduction to Stochastic Calculus Dr. Ashwin Rao Morgan Stanley, Mumbai March 11, 2011 Dr. Ashwin Rao Introduction to Stochastic Calculus
2. 2. Introduction to Stochastic Calculus Review of key concepts from Probability/Measure Theory Lebesgue Integral (Ω, F, P ) Lebesgue Integral: Ω X (ω)dP (ω) = EP X Dr. Ashwin Rao Introduction to Stochastic Calculus
3. 3. Introduction to Stochastic Calculus Review of key concepts from Probability/Measure Theory Change of measure Random variable Z with EP Z = 1 Deﬁne probability Q (A) = A Z (ω)dP (ω) ∀A ∈ F EQ X = EP [XZ ] EQ Y = EP Y Z Dr. Ashwin Rao Introduction to Stochastic Calculus
4. 4. Introduction to Stochastic Calculus Review of key concepts from Probability/Measure Theory Radon-Nikodym derivative Equivalence of measures P and Q: ∀A ∈ F, P (A) = 0 iﬀ Q (A) = 0 if P and Q are equivalent, ∃Z such that EP Z = 1 and Q (A) = A Z (ω)dP (ω) ∀A ∈ F Z is called the Radon-Nikodym derivative of Q w.r.t. P and denoted Z = dQdP Dr. Ashwin Rao Introduction to Stochastic Calculus
5. 5. Introduction to Stochastic Calculus Review of key concepts from Probability/Measure Theory Simpliﬁed Girsanov’s Theorem X = N (0, 1) θ2 Z (ω) = e θX (ω)− 2 ∀ω ∈ Ω Ep Z = 1 ∀A ∈ F , Q = ZdP A EQ X = EP [XZ ] = θ Dr. Ashwin Rao Introduction to Stochastic Calculus
6. 6. Introduction to Stochastic Calculus Information and σ-Alebgras Finite Example Set with n elements {a1 , . . . , an } Step i: consider all subsets of {a1 , . . . , ai } and its complements At step i, we have 2i +1 elements ∀i, Fi ⊂ Fi +1 Dr. Ashwin Rao Introduction to Stochastic Calculus
7. 7. Introduction to Stochastic Calculus Information and σ-Alebgras Uncountable example Fi = Information available after ﬁrst i coin tosses i Size of Fi = 22 elements Fi has 2i ”atoms” Dr. Ashwin Rao Introduction to Stochastic Calculus
8. 8. Introduction to Stochastic Calculus Information and σ-Alebgras Stochastic Process Example Ω = set of continuous functions f deﬁned on [0, T ] with f (0) = 0 FT = set of all subsets of Ω Ft : elements of Ft can be described only by constraining function values from [0, t ] Dr. Ashwin Rao Introduction to Stochastic Calculus
9. 9. Introduction to Stochastic Calculus Information and σ-Alebgras Filtration and Adaptation Filtration: ∀t ∈ [0, T ], σ-Algebra Ft . foralls t, Fs ⊂ Ft σ-Algebra σ(X ) generated by a random var X = {ω ∈ Ω|X (ω) ∈ B } where B ranges over all Borel sets. X is G-measurable if σ(X ) ⊂ G A collection of random vars X (t ) indexed by t ∈ [0, T ] is called an adapted stochastic process if ∀t, X (t ) is Ft -measurable. Dr. Ashwin Rao Introduction to Stochastic Calculus
10. 10. Introduction to Stochastic Calculus Information and σ-Alebgras Multiple random variables and Independence σ-Algebras F and G are independent if P (A ∩ B ) = P (A) · P (B ) ∀A ∈ F , B ∈ G Independence of random variables, independence of a random variable and a σ-Algebra Joint density fX ,Y (x, y ) = P ({ω|X (ω) = x, Y (ω) = y }) ∞ Marginal density fX (x ) = P ({ω|X (ω) = x }) = −∞ fX ,Y (x, y )dy X , Y independent implies fX ,Y (x, y ) = fX (x ) · fY (y ) and E [XY ] = E [X ]E [Y ] Covariance(X , Y ) = E [(X − E [X ])(Y − E [Y ])] Covariance (X ,Y ) Correlation pX ,Y = √ Varaince (X )Variance (Y ) 1 −1 1 (x −µ)T Multivariate normal density fX (x ) = √ ¯ ¯ e − 2 (x −µ)C ¯ ¯ ¯ ¯ (2π)n det (C ) X , Y normal with correlation ρ. Create independent normal variables as a linear combination of X , Y Dr. Ashwin Rao Introduction to Stochastic Calculus
11. 11. Introduction to Stochastic Calculus Conditional Expectation E [X |G] is G-measurable E [X |G](ω)dP (ω) = X (ω)dP (ω)∀A ∈ G A A Dr. Ashwin Rao Introduction to Stochastic Calculus
12. 12. Introduction to Stochastic Calculus An important Theorem G a sub-σ-Algebra of F X1 , . . . , Xm are G-measurable Y1 , . . . , Yn are independent of G E [f (X1 , . . . , Xm , Y1 , . . . , Yn )|G] = g (X1 , . . . , Xm ) How do we evaluate this conditional expectation ? Treat X1 , . . . , Xm as constants Y1 , . . . , Yn should be integrated out since they don’t care about G g (x1 , . . . , xm ) = E [f (x1 , . . . , xm , Y1 , . . . , Yn )] Dr. Ashwin Rao Introduction to Stochastic Calculus
13. 13. Introduction to Stochastic Calculus Quadratic Variation and Brownian Motion Random Walk At step i, random variable Xi = 1 or -1 with equal probability i Mi = Xj j =1 The process Mn , n = 0, 1, 2, . . . is called the symmetric random walk 3 basic observations to make about the ”increments” Independent increments: for any i0 < i1 < . . . < in , (Mi1 − Mi0 ), (Mi2 − Mi1 ), . . . (Min − Min−1 ) are independent Each incerement has expected value of 0 Each increment has a variance = number of steps (i.e., variance of 1 per step) Dr. Ashwin Rao Introduction to Stochastic Calculus
14. 14. Introduction to Stochastic Calculus Quadratic Variation and Brownian Motion Two key properties of the random walk Martingale: E [Mi |Fj ] = Mj i Quadratic Variation: [M, M ]i = j =1 (Mj − Mj −1 )2 = i Don’t confuse quadratic variation with variance of the process Mi . Dr. Ashwin Rao Introduction to Stochastic Calculus
15. 15. Introduction to Stochastic Calculus Quadratic Variation and Brownian Motion Scaled Random Walk We speed up time and scale down the step size of a random walk 1 For a ﬁxed positive integer n, deﬁne W (n )(t ) = √ n Mnt Usual properties: independent increments with mean 0 and variance of 1 per unit of time t Show that this is a martingale and has quadratic variation As n → ∞, scaled random walk becomes brownian motion (proof by central limit theorem) Dr. Ashwin Rao Introduction to Stochastic Calculus
16. 16. Introduction to Stochastic Calculus Quadratic Variation and Brownian Motion Brownian Motion Deﬁnition of Brownian motion W (t ). W ( 0) = 0 For each ω ∈ Ω, W (t ) is a continuous function of time t. independent increments that are normally distributed with mean 0 and variance of 1 per unit of time. Dr. Ashwin Rao Introduction to Stochastic Calculus
17. 17. Introduction to Stochastic Calculus Quadratic Variation and Brownian Motion Key concepts Joint distribution of brownian motion at speciﬁc times Martingale property Derivative w.r.t. time is almost always undeﬁned Quadratic variation (dW · dW = dt) dW · dt = 0, dt · dt = 0 Dr. Ashwin Rao Introduction to Stochastic Calculus
18. 18. Introduction to Stochastic Calculus Ito Calculus Ito’s Integral T I (T ) = ∆(t )dW (t ) 0 Remember that Brownian motion cannot be diﬀerentiated w.r.t time T Therefore, we cannot write I (T ) as 0 ∆(t )W (t )dt Dr. Ashwin Rao Introduction to Stochastic Calculus
19. 19. Introduction to Stochastic Calculus Ito Calculus Simple Integrands T ∆(t )dW (t ) 0 Let Π = {t0 , t1 , . . . , tn } be a partition of [0, t ] Assume ∆(t ) is constant in t in each subinterval [tj , tj +1 ] t k −1 I (t ) = ∆(u )dW (u ) = ∆(tj )[W (tj +1 )−W (tj )]+∆(tk )[W (t )−W (tk )] 0 j =0 Dr. Ashwin Rao Introduction to Stochastic Calculus
20. 20. Introduction to Stochastic Calculus Ito Calculus Properties of the Ito Integral I (t ) is a martingale t 2 Ito Isometry: E [I 2 (t )] = E [ 0 ∆ (u )du ] t 2 Quadratic Variation: [I , I ](t ) = 0 ∆ (u )du General Integrands T An example: 0 W (t )dW (t ) Dr. Ashwin Rao Introduction to Stochastic Calculus
21. 21. Introduction to Stochastic Calculus Ito Calculus Ito’s Formula T f (T , W (T )) = f (0, W (0)) + ft (t, W (t ))dt 0 T T 1 + fx (t, W (t ))dW (t ) + fxx (t, W (t ))dt 0 2 0 t t Ito Process: X (t ) = X (0) + 0 ∆(u )dW (u ) + 0 Θ(u )dW (u ) t 2 Quadratic variation [X , X ](t ) = 0 ∆ (u )du Dr. Ashwin Rao Introduction to Stochastic Calculus
22. 22. Introduction to Stochastic Calculus Ito Calculus Ito’s Formula T T f (T , X (T )) = f (0, X (0)) + ft (t, X (t ))dt + fx (t, X (t ))dX (t ) 0 0 T 1 + fxx (t, X (t ))d [X , X ](t ) 2 0 T T = f (0, X (0)) + ft (t, X (t ))dt + fx (t, X (t ))∆(t )dW (t ) 0 0 T T 1 + fx (t, X (t ))Θ(t )dt + fxx (t, X (t ))∆2 (t )dt 0 2 0 Dr. Ashwin Rao Introduction to Stochastic Calculus