Martingale

In the mathematical theory of stochastic processes, a martingale is, roughly speaking, a stochastic process that can be viewed as an abstraction of the notion of a "fair game". In a fair betting game, it is expected that on average one's winnings into the future from any point in time, given the current winnings, should remain unchanged, i.e., on average it should not be possible to improve upon nor decrease one's current winnings as the game continues, otherwise it is biased towards players either making profit or losses in the game.

Probably two of the most basic, yet important, examples of (continuous-time) martingales (with respect to an appropriate choice of filtration) are the Wiener process (otherwise known as Brownian motion) and the Poisson process. It is with respect to these two processes that stochastic integration was initially studied and understood until later, as the theory developed, it was realized that the crucial underlying property that makes stochastic integration work was the fact that the stochastic process being integrated against is a martingale. This observation made it possible to extend the stochastic integration theory from Wiener and Poisson processes to more general processes that were martingales and, more generally, semi-martingales. This played an important role in the development of stochastic calculus, a calculus that finds important applications in fields as diverse as physics, engineering, medicine and finance.

Definition of a martingale

Let $\scriptstyle (\Omega ,{\mathcal {F}},P)$ be a complete probability space. Let $\scriptstyle 0\,\in \,T\,\subset \,[0,\infty )$ and $\scriptstyle \{{\mathcal {F}}_{t}\}_{t\in T}$ be an increasing family of sub-$\scriptstyle \sigma$-algebras of $\scriptstyle {\mathcal {F}}$ (i.e., $\scriptstyle {\mathcal {F}}_{s}\,\subset \,{\mathcal {F}}_{t}\;\forall s\leq t$ ). A stochastic process $\scriptstyle \{X_{t}\}_{t\in T}$ is said to be a martingale if $\scriptstyle X_{t}$ is ${\mathcal {F}}_{t}$-measurable for each $\scriptstyle t\,\in \,T$ and it satisfies:

$E[X_{t}|{\mathcal {F}}_{s}]=X_{s}\;\forall 0\leq s\leq t,\,s,t\in T.$

The family $\scriptstyle \{{\mathcal {F}}_{t}\}_{t\in T}$ is called a filtration of $\scriptstyle {\mathcal {F}}$. Note that the property that a process is a martingale is defined with respect to the choice of filtration. If the choice of filtration is changed, then under the new filtration the process may no longer be a martingale.

Often it is also imposed that $\scriptstyle \{{\mathcal {F}}_{t}\}_{t\in T}$ satisfies the so-called usual conditions:

1. $\scriptstyle {\mathcal {F}}_{0}$ contains all the null-sets of $\scriptstyle P$
2. The filtration is right continuous: $\scriptstyle {\mathcal {F}}_{t}\,=\,\cap _{s>t}{\mathcal {F}}_{s}$

The enforcement of the usual conditions on the filtration makes it possible to choose a version of a martingale that have particularly nice sample paths which are continuous on the right and with limits on the left (abbreviated as corlol in English and càdlàg in the French). Corlol martingales are especially important due to the property that they are separable.