martingale
Martingales^{} definition
Definition. Let $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in \mathbb{T}},\mathbb{P})$ 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
$$ 
and a supermartigale if
$$ 
A submartingale that is also a supermartingale is called a martingale, i.e., a martingale satisfies
$$ 
Similarly, if the $\{{\mathcal{F}}_{t}\}$ form a decreasing collection^{} of $\sigma $subalgebras of $\mathcal{F}$, then $X$ is called a reverse submartingale if
$$ 
and a reverse supermartingale if
$$ 
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
$$ and similarly for the other definitions. This is immediate from the definition of conditional expectation.
Title  martingale 
Canonical name  Martingale 
Date of creation  20130322 13:33:09 
Last modified on  20130322 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 