martingaleNotes

martingaleNotes - IEOR 4106 Introduction to Operations...

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Unformatted text preview: IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2009, Professor Whitt Supplementary Notes on Martingales and Brownian Motion April 28-30, 2009 1. Martingales We start by defining a martingale, working in discrete time. Definition 0.1 Let { X n : n ≥ } and { Y n : n ≥ } be stochastic processes ( sequences of random variables ) . We say that { X n : n ≥ } is a martingale with respect to { Y n : n ≥ } if ( i ) E [ | X n | ] < ∞ for all n ≥ and ( ii ) E [ X n +1 | Y ,Y 1 ,...,Y n ] = X n for all n ≥ . a. More on Definition 0.1. In Definition 0.1 we think of the stochastic process { Y n : n ≥ } constituting the history or information . Then { Y k : 0 ≤ k ≤ n } is the history up to (and including) time n . The random variables Y k could be random vectors, as we illustrate below. We simply say that { X n : n ≥ } is a martingale if { X n : n ≥ } is a martingale with respect to { X n : n ≥ } ; i.e., if the history process { Y n : n ≥ } is the stochastic process { X n : n ≥ } itself. We then also say that { X n : n ≥ } is a martingale with respect to its internal history (the history generated by { X n : n ≥ } ). In the literature on martingales, the histories are usually characterized via sigma-fields of events, denoted by F n for n ≥ 0. We know whether or not each of the events in F n occurred by time n . We then write instead of (ii) above: ( ii ) E [ X n +1 |F n ] = X n for all n ≥ , where F n is understood to be the history up to time n . With that notation, we assume the history is cumulative, starting at time 0. Then F n can be understood to be shorthand for { Y k : 0 ≤ k ≤ n } . b. Conditional Expectation In order to understand the definitions above, we need to understand conditional expec- tation. The basic concepts are reviewed in the first four sections of Chapter 3 in Ross. In particular, we need to know what E [ X | Y ] means for random variables or random vectors X and Y . For this, see p. 106 of Ross. By E [ X | Y ], we mean a random variable. In particular, E [ X | Y ] = E [ X | Y = y ] when Y = y . Thus E [ X | Y ] can be regarded as a deterministic function of the random variable Y , which makes it itself be a random variable. Since (in the discrete case) E [ X ] = X y E [ X | Y = y ] P ( Y = y ) = E [ E [ X | Y ]] , we have the fundamental relation E [ E [ X | Y ]] = E [ X ] for all random variables X and Y . As a consequence, for a martingale { X n : n ≥ } with respect to { Y n : n ≥ } , we have E [ X n +1 ] = E [ E [ X n +1 | Y ,Y 1 ,...,Y n ]] = E [ X n ] for all n ≥ . Thus, by mathematical induction, for a martingale E [ X n ] = E [ X ] for all n ≥ 1. This last expected-value relation is a consequence of the martingale property, but it is not equivalent; the martingale property implies more than that....
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martingaleNotes - IEOR 4106 Introduction to Operations...

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