Stat150_Spring08_martingales

Stat150_Spring08_martingales - Statistics 150: Spring 2007...

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Unformatted text preview: Statistics 150: Spring 2007 April 24, 2007 0-1 1 Introduction Definition 1.1. A sequence Y = { Y n : n } is a martingale with respect to the sequence X = { X n : n } if, for all n , 1. E | Y n | < 2. E ( Y n +1 | X ,X 1 ,...,X n ) = Y n . Example 1.2 (Simple random walk). Let X i be i.i.d. random variables such that X i = 1 with probability p and X i =- 1 with probability q = 1- p . Then S n = X 1 + X 2 + + X n satisfies E | S n | n and E ( S n +1 | X 1 ,X 2 ,...,X n ) = S n + ( p- q ) , and Y n = S n- n ( p- q ) defines a martingale with respect to X . 1 Example 1.3 (De Moivres martingale). A simple random walk on the set { , 1 , 2 ,...,N } stops when it first hits either of the absorbing barriers at and at N ; what is the probability that it stops at the barrier ? Write X 1 ,X 2 ,..., for the steps of the walk, and S n for the position after n steps, where S = k . Define Y n = ( q/p ) S n where p = P ( X i = 1) , p + q = 1 , and < p < 1 . 2 We claim that E ( Y n +1 | X 1 ,X 2 ,...,X n ) = Y n for all n. If S n equals or N , then the process has stopped by time n and S n +1 = S n and therefore Y n +1 = Y n . If < S n < N , then E ( Y n +1 | X 1 ,X 2 ,...,X n ) = E ( ( q/p ) S n + X n +1 | X 1 ,X 2 ,...,X n ) = ( q/p ) S n [ p ( q/p ) + q ( q/p )- 1 ] = Y n , By taking expectations we see that E ( Y n +1 ) = E ( Y n ) for all n , and hence E ( Y n ) = E ( Y ) = ( q/p ) k for all n . 3 Let T be the number of steps before the absorption of the particle at either or N then E ( Y T ) = ( q/p ) k . Expanding E ( Y T ) , we have that E ( Y T ) = ( q/p ) p k + ( q/p ) N (1- p k ) where p k = P ( absorbed at | S = k ) . Therefore p k = k- N 1- N where = q/p 4 Example 1.4 (Markov chains). Let X be a discrete-time Markov chain taking values in the countable state space S with transition matrix P . Suppose that : S R is bounded and harmonic, which is to say that X j S p ij ( j ) = ( i ) for all i S. It is easily seen that Y = { ( X n ): n } is a martingale with respect to X : E ( ( X n +1 ) | X 1 ,X 2 ,...,X n ) = E ( ( X n +1 ) | X n ) = X j S p X n ,j ( j ) = ( X n ) . 5 Definition 1.5. Given a random variable Z we use the shorthand E [ Z |F n ] for E [ Z | X ,X 1 ,...X n ] . We call F = {F , F 1 ,... } a filtration . A sequence of random variables Y = { Y n : n } is said to be adapted to the filtration F if Y n is F n-measurable for all n , that is, if Y n is a function of X ,X 1 ,...,X n . 6 Definition 1.6. Let F be a filtration of the probability space ( , F , P ) , and let Y be a sequence of random variables which is adapted to F . We can rewrite our previous definition of a martingale by saying that the pair ( Y, F ) = { ( Y n , F n ): n } a martingale if for all n , 1. E | Y n | < 2. E ( Y n +1 |F n ) = Y n 7 Definition 1.7. Let F be a filtration of the probability space ( , F , P ) , and let Y be a sequence of random variables which is adapted to...
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This note was uploaded on 05/07/2008 for the course STAT 150 taught by Professor Evans during the Spring '08 term at University of California, Berkeley.

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Stat150_Spring08_martingales - Statistics 150: Spring 2007...

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