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# L12 - Lecture 12 12.1 Stochastic Integration The It...

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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 0 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 i -measurable. Definition 12.1 For a simple process h , define the Itˆo integral I ( h ) = Z T 0 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 0 Eh 2 t dt (12.4) Z T 0 ( ah t + bh 0 t ) dW t = a Z T 0 h t dW t + b Z T 0 h 0 t dW t (12.5) for any constants a, b and any simple processes h and h 0 . I ( h ) defined in ( 12.2 ) is F t k +1 -measurable. Note: It is a good exercise to show (12.3) and (12.4). In particular, you will see why is crucial to have independence between

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L12 - Lecture 12 12.1 Stochastic Integration The It...

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