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5 Stochastic Calculus Steven E Lecture Five Stochastic Calculus by 5 Stochastic Calculus Steven E. Shreve 5.1 ItˆoIntegral for a Simple Integrand Department of Mathematical Sciences 5.2 Properties for Simple Integrands Carnegie Mellon University 5.3 Construction for General Integrands 5.4 Example of an ItˆoIntegral for the 5.5 Itˆo’s Formula for One Process MAA Short Course 5.6 Solution to Exercise “Financial Mathematics” August 4-5, 2009 Portland, Oregon 1/37 2/37 The Itˆointegral problem Definition Let W be a Brownian motion defined on a probability space (Ω, F, P). A process ∆(s, ω), a function of s ≥ 0 and ω ∈ Ω, is adapted if the dependence of ∆(s, ω) on ω is as a function of the 5 Stochastic Calculus initial path fragment W (u, ω), 0 ≤ u ≤ s. In particular, ∆(s) is independent of W (t) − W (s) whenever 0 ≤ s ≤ t. 5.1 ItˆoIntegral for a Simple Integrand We want to make sense of t ∆(s) dW (s), 0 ≤ t ≤ T . Z0 Remark If g(s) is a differentiable function, then we can define t t ∆(s) dg(s)= ∆(u)g ′(s) ds. Z0 Z0 This won’t work for Brownian motion, however, because the paths of Brownian motion are not differentiable. 3/37 4/37 Simple Integrand Interpretation of Simple Integrand Let Π = {t0, t1,..., tn} be a partition of [0, T ], i.e., ◮ Think of W (s) as the price per share of an asset at time s. ◮ Think of t0, t1,..., tn−1 as the trading dates in the asset. 0= t0 ≤ t1 ≤···≤ tn = T . ◮ Think of ∆(t0), ∆(t1),..., ∆(tn−1) as the number of shares Assume that ∆(s) is constant in s on each subinterval [tk , tk+1). of the asset acquired at each trading date and held to the We call such a ∆ a simple process. next trading date. Gain from trading. I (t) = ∆(t0)[W (t) − W (t0)] = ∆(t0)W (t), 0 ≤ t ≤ t1, I (t) = ∆(t0)[W (t1) − W (t0)] + ∆(t1)[W (t) − W (t1)], t1 ≤ t ≤ t2, t1 t2 t3 t4 s I (t) = ∆(t0)[W (t1) − W (t0)] + ∆(t1)[W (t2) − W (t1)] +∆(t2)[W (t) − W (t2)], t2 ≤ t ≤ t3. One path of ∆ The process I is the Itˆointegral of the simple process ∆, i.e., Example t I (t)= ∆(s) dW (s), 0 ≤ t ≤ T . ∆(s)= W (tk ), tk ≤ s < tk+1 Z0 5/37 6/37 Expectation of Itˆointegral Theorem The Itˆointegral of a simple process has expectation zero. Proof: 5 Stochastic Calculus By definition n−1 5.2 Properties for Simple Integrands I (T )= ∆(tj ) W (tj+1) − W (tj ) . j=0 X Compute expectation term by term. Because ∆(tj ) is independent of W (tj+1) − W (tj ), we have E ∆(tj ) W (tj+1) − W (tj ) = E∆(tj ) · E W (tj+1) − W (tj ) E = ∆(tj ) · 0 = 0. 7/37 8/37 Exercise (5.1) Quadratic Variation of ItˆoIntegral Suppose Y (t), 0 ≤ t ≤ T, is a stochastic process (a function of t and ω) such that if 0 ≤ s ≤ t, then the increment Y (t) − Y (s) is Theorem The simple Itˆointegral independent of the path of Y up to time s and has expectation zero. Let {∆(s)}0≤s≤T be a simple process adapted to Y, i.e., t there is a partition Π= {t0, t1,..., tn} of [0, T ] such that ∆(s) is I (t)= ∆(u) dW (u) 0 constant in s in each subinterval [tj , tj+1), and for each s ∈ [0, T ], Z the random variable ∆(s) depends on ω only through the path of has quadratic variation Y up to time s, and hence ∆(s) is independent of Y (t) − Y (s) for T all t ∈ [s, T ]. Define the Itˆointegral [I , I ](T )= ∆2(u) du (QV ) 0 n−1 Z I (T )= ∆(t ) Y (t ) − Y (t ) . and variance j j+1 j T j 2 2 X=0 E I (T ) = E ∆ (u) du. (VAR) E (i) Show that I (T ) = 0. Z0 (ii) A simple arbitrage is a simple process ∆ such that I (T ) ≥ 0 Remark almost surely and P{I (T ) > 0} > 0. Show that there is no Both sides of (QV) are random, but the expressions in (VAR) are simple arbitrage under the assumptions of this exercise. not. (VAR) is called Itˆo’s Isometry. 9/37 10/37 Proof of (QV) Proof of (VAR) n−1 For s ∈ [tj , tj+1], we have ∆(s) = ∆(tj ) and I (T )= ∆(tj ) W (tj+1) − W (tj ) . I (s) = I (t ) + ∆(t ) W (s) − W (t ) j=0 j j j X h i Squaring and taking expectations, we obtain = I (tj ) − ∆(tj )W (tj ) + ∆(tj )W (s). n−1 h i 2 2 2 On this subinterval, quadratic variation of I comes from the E I (T ) = E ∆ (tj ) W (tj+1) − W (tj ) quadratic variation of W , which is scaled by ∆(tj ). Therefore k=j X h i 2 E . [I , I ](tj+1) − [I , I ](tj ) = ∆ (tj ) [W , W ](tj+1) − [W , W ](tj ) +2 ∆(tj )∆(tk ) W (tj+1) − W (tj ) W (tk+1) − W (tk ) j<k 2 X = ∆ (tj )(tj+1 − tj ) We use independence to simplify the pure square terms: tj+1 2 = ∆ (s) ds. 2 2 2 2 E ∆ (tj ) W (tj+1) − W (tj ) = E ∆ (tj ) · E W (tj+1) − W (tj ) Ztj tj+1 Summing over subintervals, we obtain 2 2 = E ∆ (tj ) · (tj+1 − tj )= E∆ (s) ds. n−1 T Ztj [I , I ](T )= [I , I ](t ) − [I , I ](t ) = ∆2(s) ds. j+1 j E T 2 j 0 The sum of the pure square terms is 0 ∆ (s)ds. X=0 Z 11/37 R 12/37 Proof of (VAR) (continued) It remains to show that the cross-terms have zero expectation. For j < k, the increment W (tk+1) − W (tk ) is independent of ∆(tj )∆(tk ) W (tj+1) − W (tj ) , and hence 5 Stochastic Calculus 5.3 Construction for General Integrands E ∆(tj )∆(tk ) W (tj+1) − W (tj ) W (tk+1) − W (tk ) h i = E ∆(tj )∆(tk ) W (tj+1) − W (tj ) · E W (tk+1) − W (tk ) h i = E ∆(tj )∆(tk ) W (tj+1) − W (tj ) · 0 h i = 0. 13/37 14/37 Outline of construction for general integrands Outline of construction (continued) ◮ ◮ P ∞ Given ∆(s), 0 ≤ s ≤ T , satisfying L2(Ω, F, ) is complete, and so the sequence {In(T )}n=1 has T a limit I (T ) in this space. E ∆2(s) ds < ∞, ◮ We define Z0 T ∆(s) dW (s)= I (T ) = lim In(T ). construct an approximating sequence of simple processes →∞ 0 n ∆ (s), 0 ≤ s ≤ T , such that Z n This limit does not depend on the approximating sequence T ∞ 2 {∆n}n=1. lim E ∆(s) − ∆n(s) ds = 0. n→∞ ◮ By choosing approximating sequences that converge rapidly, Z0 we can in fact make the convergence of In(T ) to I (T ) be ◮ T Set In(T )= 0 ∆n(s) dW (s). Itˆo’s isometry implies that almost sure (almost everywhere with respect to P) rather than T in L2. R 2 2 E In(T ) − Im(T )) = E ∆n(s) − ∆m(s) ds. ◮ With additional work, one can choose the approximating 0 Z sequence so that the paths of In(t), 0 ≤ t ≤ T , converge ◮ ∞ Because the sequence {∆n}n=1 converges in uniformly in t ∈ [0, T ] almost surely. This guarantees that L2(Ω × [0, T ], F ⊗ Borel([0, T ]), P × Lebesgue), it is Cauchy there is a limit I (t), 0 ≤ t ≤ T , that is a continuous function ∞ P in this space. Therefore, {In(T )}n=1 is Cauchy in L2(Ω, F, ). of t ∈ [0, T ] for P-almost every ω. 15/37 16/37 Theorem E T 2 Under the assumption [ 0 ∆ (s) ds] < ∞, the Itˆointegral tR I (t)= ∆(s) dW (s), 0 ≤ t ≤ T , Z0 is defined and continuous in t ∈ [0, T ]. We have 5 Stochastic Calculus EI (t) = 0, 0 ≤ t ≤ T . 5.4 Example of an ItˆoIntegral The quadratic variation of the Itˆointegral is t [I , I ](t)= ∆2(s) ds, 0 ≤ t ≤ T , Z0 and the Itˆointegral satisfies Itˆo’s Isometry t Var[I (t)] = E I 2(t) = E ∆2(s) ds , 0 ≤ t ≤ T . Z0 17/37 18/37 T W s dW s 0 ( ) ( ) Divide [0, T ] into n equal subintervals. Define By definition, R T jT jT (j + 1)T ∆n(s)= W for ≤ s < . W (s) dW (s) n n n Z0 n−1 jT (j + 1)T jT = lim W W − W . n→∞ n n n j X=0 jT To simplify notation, we denote Wj = W n . Then W0 = W (0) = 0, Wn = W (T ), and T n−1 W (s) dW (s) = lim Wj (Wj+1 − Wj ). n→∞ t1 t2 t3 t = T s 0 j 4 Z X=0 One path of W (s) and ∆4(s) 19/37 20/37 n−1 n−1 n−1 n−1 From the previous page, we have 1 2 1 2 1 2 (Wj+1 − Wj ) = Wj+1 − Wj Wj+1 + Wj 2 2 2 n−1 n−1 j=0 j=0 j=0 j=0 1 2 1 2 X X X X Wj (Wj+1 − Wj )= Wn − (Wj+1 − Wj ) . n n−1 n−1 2 2 1 1 j=0 j=0 = W 2 − W W + W 2 X X 2 k j j+1 2 j k=1 j=0 j=0 Letting n → ∞, we get X X X n−1 n−1 n−1 T 1 2 1 2 1 2 1 2 1 1 2 1 = W + W − Wj Wj + W W (s) dW (s)= W (T ) − [W , W ](T ) = W (T ) − T .
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