---
title: "Martingale"
firstLetter: "M"
publishDate: 2021-10-31
categories:
- Mathematics

date: 2021-10-18T10:01:46+02:00
draft: false
markup: pandoc
---

# Martingale

A **martingale** is a type of stochastic process
(i.e. a time-indexed [random variable](/know/concept/random-variable/))
with important and useful properties,
especially for stochastic calculus.

For a stochastic process $\{ M_t : t \ge 0 \}$
on a probability space $(\Omega, \mathcal{F}, P)$ with filtration $\{ \mathcal{F}_t \}$
(see [$\sigma$-algebra](/know/concept/sigma-algebra/)),
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 essentially means that a martingale is an unbiased random walk.
Accordingly, the [Wiener process](/know/concept/wiener-process/) $\{ B_t \}$
(Brownian motion) is a prime example of a martingale
(with respect to its own filtration),
since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition.

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.