Categories: Mathematics, Stochastic analysis.

Martingale

A martingale is a type of stochastic process with important and useful properties, especially for stochastic calculus.

For a stochastic process $\{ M_t : t \ge 0 \}$ on a probability filtered space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$, then $M_t$ is a martingale if it satisfies all of the following:

1. $M_t$ is $\mathcal{F}_t$-adapted, meaning the filtration $\mathcal{F}_t$ contains enough information to reconstruct the current and all past values of $M_t$.
2. For all times $t \ge 0$, the expectation value exists $\mathbf{E}(M_t) < \infty$.
3. For all $s, t$ satisfying $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) = M_s$, meaning the increment $M_t \!-\! M_s$ is always expected to be zero $\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$.

The last condition is called the martingale property, and basically means that a martingale is an unbiased random walk. Accordingly, the Wiener process $B_t$ (Brownian motion) is an example of a martingale, since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition.

Martingales are easily confused with Markov processes, because stochastic processes will often be both, e.g. the Wiener process. However, these are distinct concepts: the martingale property says nothing about history-dependence, and the Markov property does not say what the future expectation should be.

Modifying property (3) leads to two common generalizations. The stochastic process $M_t$ above is a submartingale if the current value is a lower bound for the expectation:

1. For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$.

Analogouly, $M_t$ is a supermartingale if the current value is an upper bound instead:

1. For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$.

Clearly, submartingales and supermartingales are biased random walks, since they will tend to increase and decrease with time, respectively.

References

1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.