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<t\in{\mathbb R}_+}.

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