# 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

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

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