local properties of processes

Many properties of stochastic processesMathworldPlanetmath, such as the martingaleMathworldPlanetmath property, can be generalized to a corresponding local property. The local properties can be more useful than the original property because they are often preserved under certain transformationsMathworldPlanetmath of processes, such as random time changes and stochastic integration.

Let (Ξ©,β„±,(β„±t)tβˆˆπ•‹,β„™) be a filtered probability space and Ο€ be a property of stochastic processes with time index setMathworldPlanetmathPlanetmath π•‹βŠ†β„. The property Ο€ is said to hold locally for a process X if there exists a sequence of stopping times (Ο„n)nβˆˆβ„€+ taking values in 𝕋βˆͺ{∞} and almost surely increasing to infinityMathworldPlanetmath, such that the stopped processes XΟ„n have property Ο€ for each n.

Often, the index set 𝕋 has a minimal element t0, in which case it is convenient to extend the concept of localization slightly so that Ο€ holds locally if there is a sequence of stopping times Ο„n almost surely increasing to infinity and such that 1{Ο„n>t0}⁒XΟ„n have property Ο€.

The property of locally satisfying Ο€ is often denoted as Ο€loc. Similarly, if Ο€ is a class of processes then the processes which are locally in Ο€ is denoted by Ο€loc. Letting Ο„n be the stopping times taking the constant value ∞ shows that every process in Ο€ is also locally in Ο€, so Ο€βŠ†Ο€loc.

In most cases where localization is used, such as with the class of right-continuous martingales, for any process X in Ο€ and stopping time Ο„ then 1{Ο„>t0}⁒XΟ„ is also in Ο€. If this is the case then it is easily shown that a process is locally in Ο€loc if and only if it is locally in Ο€. So, (Ο€loc)loc=Ο€loc.

Examples of commonly used local properties are as follows.

  1. 1.

    A process X is said to be a local martingale if it is locally a right-continuous martingale. That is, if there is a sequence of stopping times Ο„n almost surely increasing to infinity and such that 1{Ο„n>t0}⁒XΟ„n∧t is integrable and,


    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 intervalMathworldPlanetmathPlanetmath 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 integral, but the martingale property is not. Examples of local martingales which are not proper martingales are given by solutions to the stochastic differential equation


    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.

  2. 2.

    A local submartingale (resp. local supermartingale) is a right-continuous process which is locally a submartingale (resp. supermartingale). A local submartingale can be shown to be a submartingale if and only if Xt0 is integrable and the set {XΟ„βˆ¨0:τ≀t⁒ is a stopping time} is locally integrable for every tβˆˆπ•‹. In particular, every nonpositive local submartingale X for which Xt0 is integrable is a submartingale. Similarly every nonnegative supermartingale X such that Xt0 is integrable is a supermartingale.

  3. 3.

    An increasing and non-negative process X is locally integrable if it is locally an integrable process. That is, there is a sequence of stopping times Ο„n increasing to infinity and such that 𝔼⁒[1{Ο„n>t0}⁒|XΟ„n∧t|]<∞ for every nβˆˆβ„€+ and tβˆˆπ•‹. By monotonicity of X, this is equivalent to 𝔼⁒[1{Ο„n>t0}⁒|XΟ„n|]<∞. For example, the maximum process Xt*≑sups≀t⁑|Xs| of a local martingale X is locally integrable.

  4. 4.

    A process X is said to be locally bounded if there is a sequence of stopping times Ο„n almost surely increasing to infinity and such that 1{Ο„nβ‰₯t0}⁒XΟ„n are uniformly bounded processes. For example, in discrete-time so 𝕋=β„€+, then every predictable process is locally bounded.

Similarly, in continuous-time, if Ο€ is a property of stochastic processes and X is a stochastic process such that the left limits of Xt with respect to t exist everywhere, then X is said to prelocally satisfy Ο€ if there is a sequence of stopping times Ο„n almost surely increasing to infinity and such that the prestopped processes 1{Ο„n>t0}⁒XΟ„n- satisfy Ο€.

Title local properties of processes
Canonical name LocalPropertiesOfProcesses
Date of creation 2013-03-22 18:38:50
Last modified on 2013-03-22 18:38:50
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Definition
Classification msc 60G05
Classification msc 60G40
Classification msc 60G48
Related topic LocalMartingale
Defines local property
Defines local submartingale
Defines local martingale
Defines locally integrable process
Defines locally bounded process
Defines prelocalization
Defines prelocal property