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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 i-measurable. 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 ....
View Full Document

This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online