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:

- $M_t$ 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.

## References

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