IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt Topics for Discussion: Thursday, December 7, 2006 More on Martingales and Brownian Motion

a. More on Definition 0.1. In Definition 0.1 we think of the stochastic process {Yn : n ≥ 0} constituting the history or information. Then {Yk : 0 ≤ k ≤ n} is the history up to (and including) time n. The random variables Yk could be random vectors, as we illustrate below. We simply say that {Xn : n ≥ 0} is a martingale if {Xn : n ≥ 0} is a martingale with respect to {Xn : n ≥ 0}; i.e., if the history process {Yn : n ≥ 0} is the stochastic process {Xn : n ≥ 0} itself. We then also say that {Xn : n ≥ 0} is a martingale with respect to its internal history (the history generated by {Xn : n ≥ 0}). In the literature on martingales, the histories are usually characterized via sigma-fields of events, denoted by Fn for n ≥ 0. We know whether or not each of the events in Fn occurred by time n. We then write instead of (ii) above: