---
title: "Martingale"
date: 2021-10-31
categories:
- Mathematics
- Stochastic analysis
layout: "concept"
---

A **martingale** is a type of
[stochastic process](/know/concept/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](/know/concept/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](/know/concept/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](/know/concept/markov-process/),
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:

3.  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:

3.  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.