Categories: Mathematics, Stochastic analysis.

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:

- \(M_t\) is \(\mathcal{F}_t\)-adapted, meaning the filtration \(\mathcal{F}_t\) contains enough information to reconstruct the current and all past values of \(M_t\).
- For all times \(t \ge 0\), the expectation value exists \(\mathbf{E}(M_t) < \infty\).
- 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:

- 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:

- 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.

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

© Marcus R.A. Newman, a.k.a. "Prefetch".
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