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09StochChpater3

09StochChpater3 - Chapter 3 Stochastic Integration and Ito...

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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 vector-valued or a Banach space-valued 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 0 f ( t ) dB ( t ), where f ( t ) is a real-valued 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 = 0 , 1 , · · · , n - 1. We write Δ B i = B ( t i +1 ) - B ( t i ), and define I ( f ) = Z T 0 f ( t ) dB ( t ) = n - 1 X i =0 c i Δ B i . (3.1.1) 83

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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 0 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 0 and variance σ 2 = E (( I ( f )) 2 ) = Z T 0 | 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 ( B i ) 2 ) = X i c 2 i ( t i +1 - t i ) = Z T 0 | 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 0 | 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 0 and variance σ 2 = k f k 2 = R T 0 | 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 0 f ( t ) g ( t ) dt . Proof . This follows from E ( ( I ( f ) + I ( g )) 2 ) = Z T 0 | f + g | 2 = Z T 0 | f | 2 + 2 Z T 0 fg + Z T 0 | g | 2 and also E ( ( I ( f ) + I ( g )) 2 ) = E ( | I ( f ) | 2 + 2 I ( f ) I ( g ) + | I ( g ) | 2 ) = Z T 0 | f | 2 + 2 E ( I ( f ) I ( g )) + Z T 0 | g | 2 . / Next we want to consider the relationship of ( R T 0 f ( t ) dB ( t ))( ω ) and R T 0 f ( t ) dB ( t, ω ). Note the B ( t, ω ) has unbounded variation (but has finite quadratic variation), R T 0 f ( t ) dB ( t, ω ) is not defined as a Riemann Stieltjes integral literally. We redefine the integral as follows: For [ a, b ] [0 , T ] (RS) Z b a f ( t ) dB ( t, ω ) := h f ( t ) B ( t, ω ) i b a - Z b a B ( t, ω ) df ( t ) provided the last term is defined.

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86 CHAPTER 3.
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