Martingales

Abstract

We consider a family of processes called martingales. Though a number of other illustrative examples will be given, the most immediate interpretation is that of a fair game. The discrete parameter case is considered for simplicity. The parameter n is assumed to run over all the integers, the positive integers or the negative integers. Let X n be a sequence of random variables with B n a corresponding nondecreasing sequence of Borei subfields of F. The random variable X n is assumed to be measurable with respect to B n. In fact, B n is often taken to be the Borei field generated by X k , k ≤ n, though this is not necessarily the case in our discussion. Also let E[|X n|] < ∞ for each n. The sequence {X n} is called a martingale with respect to the Borel fields {B n} if $$ {X_m} = E\left[ {{X_n}|{\mathcal{B}_m}} \right],\quad m < n, $$ ((1)) almost surely. Think of a sequence of gambles at the times n. B n can be thought of as corresponding to the information available to the player at time n. X n is the cumulative gain (or loss) of the player up to and including time n. Condition (1) then states that the sequence of plays is “fair” in the sense that the conditional net gain from time m to n given information up to and including time m is zero. If the condition (1) is replaced by $$ {X_m} \geqslant E\left[ {{X_n}|{\mathcal{B}_n}} \right],\quad m < n, $$ ((2)) almost surely, the sequence {X n } is called a supermartingale relative to{B n }.