7 - Copyright c 2009 by Karl Sigman 1 Introduction to...

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Unformatted text preview: Copyright c 2009 by Karl Sigman 1 Introduction to Martingales in discrete time Martingales are stochastic processes that are meant to capture the notion of a fair game in the context of gambling. In a fair game, each gamble on average, regardless of the past gam- bles, yields no profit or loss. But the reader should not think that martingales are used just for gambling; they pop up naturally in numerous applications of stochastic modeling. They have enough structure to allow for strong general results while also allowing for dependencies among variables. Thus they deserve the kind of attention that Markov chains do. Gambling, however, supplies us with insight and intuition through which a great deal of the theory can be understood. 1.1 Basic definitions and examples Definition 1.1 A stochastic process X = { X n : n } is called a martingale (MG) if C1: E ( | X n | ) < , n , and C2: E ( X n +1 | X ,...,X n ) = X n , n . Notice that property C2 can equivalently be stated as E ( X n +1- X n | X ,...,X n ) = 0 , n . (1) In the context of gambling, by letting X n denote your total fortune after the n th gamble, this then captures the notion of a fair game in that on each gamble, regardless of the outcome of past gambles, your expected change in fortune is 0; on average you neither win or lose any money. Taking expected values in C2 yields E ( X n +1 ) = E ( X n ) , n 0, and we conclude that E ( X n ) = E ( X ) , n , for any MG; At any time n , your expected fortune is the same as it was initially. For notational simplicity, we shall let G n = { X ,...,X n } denote all the events determined by the rvs X ,...,X n , and refer to it as the information determined by X up to and including time n . Note that G n G n +1 , n 0; information increases as time n increases. Then the martingale property C2 can be expressed nicely as E ( X n +1 |G n ) = X n , n . A very important fact is the following which we will make great use of throughout our study of martingales: 1 Proposition 1.1 Suppose that X is a stochastic process satisfying C1. Let G n = { X ,...,X n } , n . Suppose that F n = { U ,...,U n } , n is information for some other stochastic process such that it contains the information of X : G n F n , n . Then if E ( X n +1 |F n ) = X n , n , then in fact E ( X n +1 |G n ) = X n , n , so X is a MG. G n F n implies that F n also determines X ,...,X n , but may also determine other things as well. So the above Proposition allows us to verify condition C2 by using more information than is necessary. In many instances, this helps us verify C2 in a much simpler way than would be the case if we directly used G n ....
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7 - Copyright c 2009 by Karl Sigman 1 Introduction to...

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