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Unformatted text preview: Chapter 3 Stochastic Integration and Ito Calculus 3.1 Wiener integral The most basic integration theory is based on the Riemann integral of a real valued function on an interval [0 ,T ]. It is easy to extend the integrand to be a vectorvalued or a Banach spacevalued function. The Riemann Stielt jes integral is an extension such that the integrator is a function of bounded variation. The Lebesgue integral is to allow the integrator to be a measure on a more general measure space. In this section, we will consider the Wiener integral R T f ( t ) dB ( t ), where f ( t ) is a realvalued function, integrating over the Brownian motion. Suppose f is a step function on [0 ,T ], f ( t ) = c i t i ≤ t < t i +1 , i = , 1 , ··· ,n 1. We write Δ B i = B ( t i +1 ) B ( t i ), and define I ( f ) = Z T f ( t ) dB ( t ) = n 1 X i =0 c i Δ B i . (3.1.1) 83 84 CHAPTER 3. STOCHASTIC INTEGRATION AND ITO CALCULUS Example 3.1.1 . Let f ( t ) be a step function on [0 , 3] that takes values 1 , 1 , 2 on the intervals [0 , 1), [1 , 2) and [2 , 3] respectively. Then I ( f ) = Z 3 f ( t ) dB ( t ) = ( 1)( B (1) B (0)) + 1 · ( B (2) B (1)) + 2 · ( B (3) B (2)) = B (1) + ( B (2) B (1)) + 2 · ( B (3) B (1)) ∼ N (0 , 1) + N (0 , 1) + N (0 , 4) (independent sum) = N (0 , 6) . Lemma 3.1.1. Let f be a step function on [0,T]. Then I ( f ) ∈ L 2 (Ω) , it is a normal r.v. with mean and variance σ 2 = E (( I ( f )) 2 ) = Z T  f ( t )  2 dt . Proof . Recall that if X 1 , ··· ,X n are independent and X i ∼ N ( μ i ,σ 2 i ), then a 1 X 1 + ··· + a n X n ∼ N ( a 1 μ 1 + ··· + a n μ n , a 2 1 σ 2 1 + ··· + a 2 n σ 2 n ) . It follows that I ( f ) is a normal r.v. with mean 0. For the variance, we have E ( I ( f ) 2 ) = E ( X i,j c i c j Δ B i Δ B j ) = X i c 2 i E ( (Δ B i ) 2 ) = X i c 2 i ( t i +1 t i ) = Z T  f ( t )  2 dt . / Lemma 3.1.2. Suppose f ∈ L 2 [0 ,T ] , then there exist a sequence of step func tion { f n } converges to f a.e., and { I ( f n ) } ∞ n =1 is a Cauchy sequence in L 2 (Ω) . Proof . The first statement is well known. The second statement follows from E ( ( I ( f n ) I ( f m )) 2 ) = Z T  f n ( t ) f m ( t )  2 dt . / 3.1. WIENER INTEGRAL 85 Definition 3.1.3. For f ∈ L 2 [0 ,T ] , we define I ( f ) = lim n →∞ I ( f n ) where { f n } is as in Lemma 3.1.2, and call I ( f ) the Wiener integral of f . Clearly I ( f ) ∈ L 2 (Ω) by the completeness of L 2 (Ω) and the definition is independent of the choice of the subsequence { f n } . Proposition 3.1.4. For f ∈ L 2 [0 ,T ] , I ( f ) ∈ L 2 (Ω) is a normal r.v. with mean and variance σ 2 = k f k 2 = R T  f ( t )  2 dt . Proof . We need to use the fact that if X n ∼ N ( μ n ,σ n ) and if X n → X in probability (or in L 2 (Ω)), then X ∼ N ( μ,σ ) with μ = lim n →∞ μ n and σ = lim n →∞ σ n / Corollary 3.1.5. If f,g ∈ L 2 [0 ,T ] , then E ( I ( f ) I ( g )) = R T f ( t ) g ( t ) dt ....
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This note was uploaded on 05/23/2010 for the course MATH 987 taught by Professor Cheung,cecilia during the Spring '08 term at CUHK.
 Spring '08
 CHEUNG,CECILIA
 Calculus

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