MartingalesMathworldPlanetmath definition

Definition. Let (Ω,,(t)t𝕋,) be a filtered probability space and (Xt) be a stochastic processMathworldPlanetmath such that Xt is integrable ( for all t𝕋. Then, X=(Xt,t) is called a submartingale if

𝔼[Xt|s]Xs,for every s<t, a.e.[],

and a supermartigale if

𝔼[Xt|s]Xs,for every s<t, a.e.[].

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

𝔼[Xt|s]=Xs,for every s<t, a.e.[].

Similarly, if the {t} form a decreasing collectionMathworldPlanetmath of σ-subalgebras of , then X is called a reverse submartingale if

𝔼[Xs|t]Xt,for every s<t, a.e.[],

and a reverse supermartingale if

𝔼[Xs|t]Xt,for every s<t, a.e.[].


  • 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to

    AXt𝑑AXs𝑑for every As 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