local martingale


Let (Ξ©,β„±,(β„±t)tβˆˆπ•‹,β„™) be a filtered probability space, where the time index setMathworldPlanetmathPlanetmath π•‹βŠ†β„ has minimal element t0. The most common cases are discrete-time, with 𝕋=β„€+, and continuousPlanetmathPlanetmath time where 𝕋=ℝ+, in which case t0=0.

A process X is said to be a local martingalePlanetmathPlanetmath if it is locally (http://planetmath.org/LocalPropertiesOfProcesses) a right-continuous martingaleMathworldPlanetmath. That is, if there is a sequence of stopping times Ο„n almost surely increasing to infinityMathworldPlanetmath and such that the stopped processes 1{Ο„n>t0}⁒XΟ„n are martingales. Equivalently, 1{Ο„n>t0}⁒XΟ„n∧t is integrable and

1{Ο„n>t0}XΟ„n∧s=𝔼[1{Ο„n>t0}XΟ„n∧tβˆ£β„±s]

for all s<tβˆˆπ•‹. In the discrete-time case where 𝕋=β„€+ then it can be shown that a local martingale X is a martingale if and only if 𝔼⁒[|Xt|]<∞ for every tβˆˆβ„€+. More generally, in continuous-time where 𝕋 is an interval of the real numbers, then the stronger property that

{XΟ„:τ≀t⁒ is a stopping time}

is uniformly integrable for every tβˆˆπ•‹ gives a necessary and sufficient condition for a local martingale to be a martingale.

Local martingales form a very important class of processes in the theory of stochastic calculus. This is because the local martingale property is preserved by the stochastic integralMathworldPlanetmath, but the martingale property is not. Examples of local martingales which are not proper martingales are given by solutions to the stochastic differential equation

d⁒X=Xα⁒d⁒W

where X is a nonnegative process, W is a Brownian motionMathworldPlanetmath and Ξ±>1 is a fixed real number.

An alternative definition of local martingales which is sometimes used requires XΟ„n to be a martingale for each n. This definition is slightly more restrictive, and is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the definition given above together with the condition that Xt0 must be integrable.

Title local martingale
Canonical name LocalMartingale
Date of creation 2013-03-22 15:12:43
Last modified on 2013-03-22 15:12:43
Owner skubeedooo (5401)
Last modified by skubeedooo (5401)
Numerical id 8
Author skubeedooo (5401)
Entry type Definition
Classification msc 60G07
Classification msc 60G48
Related topic Martingale
Related topic LocalPropertiesOfProcesses