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Unformatted text preview: Lecture 12 Stochastic Integration 12.1 The Itˆo integral of simple processes We will define the Itˆo stochastic integral in this lecture through an approximation procedure, same as in the definition of the Riemann integral. Let W be a standard Brownian motion and {F t } t ≥ be the Brownian filtration. An adapted process h = { h t } is said to be simple if it is a random step function: h t = k X i =1 ξ i I ( t i , t i +1 ] ( t ) , (12.1) for some positive integer k , a finite sequence 0 < t 1 < ··· < t k +1 ≤ T , and random variables ξ 1 ,...,ξ k such that ξ i is F t imeasurable. Definition 12.1 For a simple process h , define the Itˆo integral I ( h ) = Z T h t dW t = k X i =1 ξ i ( W t i +1 W t i ) . (12.2) Proposition 12.1 I ( h ) satisfies the following properties: EI ( h ) = 0 (12.3) E [ I ( h )] 2 = Z T Eh 2 t dt (12.4) Z T ( ah t + bh t ) dW t = a Z T h t dW t + b Z T h t dW t (12.5) for any constants a,b and any simple processes h and h ....
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This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.
 Spring '08
 Staff

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