martingale

Martingales definition

Definition. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{T}},\mathbb{P})$ be a filtered probability space and $(X_{t})$ be a stochastic process such that $X_{t}$ is integrable (http://planetmath.org/Integral2) for all $t\in\mathbb{T}$ . Then, $X=(X_{t},\mathcal{F}_{t})$ is called a submartingale if

\mathbb{E}^{\mathbb{P}}[X_{t}|\mathcal{F}_{s}]\geq X_{s},\,\mbox{for every $s<% t$, a.e.[$\mathbb{P}$],}

and a supermartigale if

\mathbb{E}^{\mathbb{P}}[X_{t}|\mathcal{F}_{s}]\leq X_{s},\,\mbox{for every $s<% t$, a.e.[$\mathbb{P}$].}

A submartingale that is also a supermartingale is called a martingale, i.e., a martingale satisfies

\mathbb{E}^{\mathbb{P}}[X_{t}|\mathcal{F}_{s}]=X_{s},\,\mbox{for every $s<t$, a.e.[$\mathbb{P}$].}

Similarly, if the $\{\mathcal{F}_{t}\}$ form a decreasing collection of $\sigma$ -subalgebras of $\mathcal{F}$ , then $X$ is called a reverse submartingale if

\mathbb{E}^{\mathbb{P}}[X_{s}|\mathcal{F}_{t}]\geq X_{t},\,\mbox{for every $s<% t$, a.e.[$\mathbb{P}$],}

and a reverse supermartingale if

\mathbb{E}^{\mathbb{P}}[X_{s}|\mathcal{F}_{t}]\leq X_{t},\,\mbox{for every $s<% t$, a.e.[$\mathbb{P}$].}

Remarks

•

The martingale property captures the idea of a fair bet, where the expected future value is equal to the current value.
•

The submartingale property is equivalent to

$\int_{A}X_{t}\,d\mathbb{P}\geq\int_{A}X_{s}\,d\mathbb{P}\,\,\,\mbox{for every % $A\in\mathcal{F}_{s}$ and $s<t$}$

and similarly for the other definitions. This is immediate from the definition of conditional expectation.

Title	martingale
Canonical name	Martingale
Date of creation	2013-03-22 13:33:09
Last modified on	2013-03-22 13:33:09
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	25
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60G46
Classification	msc 60G44
Classification	msc 60G42
Related topic	LocalMartingale
Related topic	DoobsOptionalSamplingTheorem
Related topic	ConditionalExpectationUnderChangeOfMeasure
Related topic	MartingaleConvergenceTheorem
Defines	martingale
Defines	supermartingale
Defines	submartingale
Defines	reverse submartingale
Defines	reverse supermartingale