predictable process

A predictable process is a real-valued stochastic processMathworldPlanetmath whose values are known, in a sense, just in advance of time. Predictable processes are also called previsible.

1 predictable processes in discrete time

Suppose we have a filtrationPlanetmathPlanetmath ( (n)n+ on a measurable spaceMathworldPlanetmathPlanetmath (Ω,). Then a stochastic process Xn is predictable if Xn is n-1-measurable ( for every n1 and X0 is 0-measurable. So, the value of Xn is known at the previous time step. Compare with the definition of adapted processes for which Xn is n-measurable.

2 predictable processes in continuous time

In continuousPlanetmathPlanetmath time, the definition of predictable processes is a little more subtle. Given a filtration (t) with time index t ranging over the non-negative real numbers, the class of predictable processes forms the smallest set of real valued stochastic processes containing all left-continuous t-adapted processes and which is closed under taking limits of a sequence of processes.

Equivalently, a real-valued stochastic process


is predictable if it is measurable with respect to the predictable sigma algebra . This is defined as the smallest σ-algebra on +×Ω making all left-continuous and adapted processes measurable.

Alternatively, is generated by either of the following collectionsMathworldPlanetmath of subsets of +×Ω

=σ({(T,):T is a stopping time}{{0}×A:A0})
=σ({[T,):T is a predictable stopping time})

Note that in these definitions, the sets (T,) and [T,) are stochastic intervals, and subsets of +×Ω.

3 general predictable processes

The definition of predictable process given above can be extended to a filtration (t) with time index t lying in an arbitrary subset 𝕋 of the extended real numbers. In this case, the predictable sets form a σ-algebra on 𝕋×Ω. If 𝕋 has a minimum element t0 then let S be the collection of sets of the form {t0}×A for At0, otherwise let S be the empty setMathworldPlanetmath.Then, the predictable σ-algebra is defined by

=σ({(t,]×A:t𝕋,At}S)=σ({(T,]:T:Ω𝕋 is a stopping time}S).

Here, (t,] and (T,] are understood to be intervals containing only times in the index setMathworldPlanetmathPlanetmath 𝕋. If 𝕋 is an interval of the real numbers then can be equivalently defined as the σ-algebra generated by the class of left-continuous and adapted processes with time index ranging over 𝕋.

A stochastic process X:𝕋×Ω is predictable if it is -measurable. It can be verified that in the cases where 𝕋=+ or 𝕋=+ then this definition agrees with the ones given above.

Title predictable process
Canonical name PredictableProcess
Date of creation 2013-03-22 18:36:30
Last modified on 2013-03-22 18:36:30
Owner gel (22282)
Last modified by gel (22282)
Numerical id 12
Author gel (22282)
Entry type Definition
Classification msc 60G07
Related topic PredictableStoppingTime
Related topic ProgressivelyMeasurableProcess
Related topic OptionalProcess
Defines predictable
Defines previsible