Martingale

Reference

• Probability Theory and Examples, Rick Durrett

If you choose to stop playing at some bounded time \(N\), then your expected winnings \(EX_N\) are equal to your initial fortune \(X_0\).

Conditional Expectation

Definition

Given are a probability space \((\Omega,\mathcal{F}_o,P )\), a \(\sigma\)-field \(\mathcal{F}\in \mathcal{F}_o\), and a random variable \(X \in \mathcal{F}_o\) with \(E|X|<\infty\). We define the conditional expectation of \(X\) given \(\mathcal{F}\), denoted by \(E(X|\mathcal{F})\), to be any random variable \(Y\) that satisfies

(i) \(Y\in \mathcal{F}\), i.e. is \(\mathcal{F}\) measurable.

(ii) for all \(A\in \mathcal{F}\), \(\int_A XdP=\int_A Y dP\).

The first thing to be settled is that the conditional expectation exists and is unique.