# Martingales and Elementary Integrals

A martingale is a stochastic process which stays the same, on average. That is, the expected future value conditional on the present is equal to the current value. Examples include the wealth of a gambler as a function of time, assuming that he is playing a fair game. The canonical example of a continuous time martingale is Brownian motion and, in discrete time, a symmetric random walk is a martingale. As always, we work with respect to a filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$. A process ${X}$ is said to be integrable if the random variables ${X_t}$ are integrable, so that ${{\mathbb E}[\vert X_t\vert]<\infty}$.

Definition 1 A martingale, ${X}$, is an integrable process satisfying

 $\displaystyle X_s={\mathbb E}[X_t\mid\mathcal{F}_s]$

for all ${s.