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# lec0426 - IEOR 4106 Spring 2011 Professor Whitt Martingales...

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IEOR 4106, Spring 2011, Professor Whitt Martingales, Gambling and Brownian Motion April 26, 2011 1 Martingales We start by defining a martingale, working in discrete time. This first section is an alternative, supplement, to the notes of the last class. Definition 1.1 Let { X n : n 0 } and { Y n : n 0 } be stochastic processes ( sequences of random variables ) . We say that { X n : n 0 } is a martingale with respect to { Y n : n 0 } if ( i ) E [ | X n | ] < for all n 0 and ( ii ) E [ X n +1 | Y 0 , Y 1 , . . . , Y n ] = X n for all n 0 . a. More on Definition 1.1. In Definition 1.1 we think of the stochastic process { Y n : n 0 } 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 0 } is a martingale if { X n : n 0 } is a martingale with respect to { X n : n 0 } ; i.e., if the history process { Y n : n 0 } is the stochastic process { X n : n 0 } itself. We then also say that { X n : n 0 } is a martingale with respect to its internal history (the history generated by { X n : n 0 } ). 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 0 , 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 ]] ,

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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 0 } with respect to { Y n : n 0 } , we have E [ X n +1 ] = E [ E [ X n +1 | Y 0 , Y 1 , . . . , Y n ]] = E [ X n ] for all n 0 . Thus, by mathematical induction, for a martingale E [ X n ] = E [ X 0 ] 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. c. More on the Martingale Definition. We can now say more about Definition 1.1. First, we observe that Property (i) is a technical regularity condition, while property (ii) is the key property. It could be expressed equivalently in two parts: ( ii - a ) 0 E [ X n | Y 0 , Y 1 , . . . , Y n ] = X n for all n 0 .
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lec0426 - IEOR 4106 Spring 2011 Professor Whitt Martingales...

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