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Stat150_Spring08_martingales

Stat150_Spring08_martingales - Statistics 150 Spring 2007...

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Statistics 150: Spring 2007 April 24, 2007 0-1
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1 Introduction Definition 1.1. A sequence Y = { Y n : n 0 } is a martingale with respect to the sequence X = { X n : n 0 } if, for all n 0 , 1. E | Y n | < 2. E ( Y n +1 | X 0 , 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
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Example 1.3 (De Moivre’s martingale). A simple random walk on the set { 0 , 1 , 2 , . . . , N } stops when it first hits either of the absorbing barriers at 0 and at N ; what is the probability that it stops at the barrier 0 ? Write X 1 , X 2 , . . . , for the steps of the walk, and S n for the position after n steps, where S 0 = k . Define Y n = ( q/p ) S n where p = P ( X i = 1) , p + q = 1 , and 0 < p < 1 . 2
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We claim that E ( Y n +1 | X 1 , X 2 , . . . , X n ) = Y n for all n. If S n equals 0 or N , then the process has stopped by time n and S n +1 = S n and therefore Y n +1 = Y n . If 0 < 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 0 ) = ( q/p ) k for all n . 3
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Let T be the number of steps before the absorption of the particle at either 0 or N then E ( Y T ) = ( q/p ) k . Expanding E ( Y T ) , we have that E ( Y T ) = ( q/p ) 0 p k + ( q/p ) N (1 - p k ) where p k = P ( absorbed at 0 | S 0 = k ) . Therefore p k = ρ k - ρ N 1 - ρ N where ρ = q/p 4
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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 φ : S R is bounded and harmonic, which is to say that j S p ij φ ( j ) = φ ( i ) for all i S. It is easily seen that Y = { φ ( X n ): n 0 } is a martingale with respect to X : E ( φ ( X n +1 ) | X 1 , X 2 , . . . , X n ) = E ( φ ( X n +1 ) | X n ) = j S p X n ,j φ ( j ) = φ ( X n ) . 5
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Definition 1.5. Given a random variable Z we use the shorthand E [ Z |F n ] for E [ Z | X 0 , X 1 , . . . X n ] . We call F = {F 0 , F 1 , . . . } a filtration . A sequence of random variables Y = { Y n : n 0 } 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 0 , X 1 , . . . , X n . 6
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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 0 } a martingale if for all n 0 , 1. E | Y n | < 2. E ( Y n +1 |F n ) = Y n 7
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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 F . We call the pair ( Y, F ) a submartingale if, for all n 0 , 1. E [ Y + n ] < 2. E ( Y n +1 |F n ) Y n or a supermartingale if, for all n 0 , 3. E [ Y - n ] < 4. E ( Y n +1 |F n ) Y n 8
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We call the pair ( S, F ) predictable if S n is F n - 1 -measurable for all n 1 . We call a predictable process ( S, F ) increasing if S 0 = 0 and P ( S n S n +1 ) = 1 for all n .
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