simple predictable process

Simple predictable processes are a particularly simple class of stochastic processesMathworldPlanetmath, for which the Ito integral can be defined directly. They are often used as the starting point for defining stochastic integrals of more general predictable integrands.

Suppose we are given a filtrationPlanetmathPlanetmath ( (t) on the measurable spaceMathworldPlanetmathPlanetmath (Ω,) with time index t ranging over the nonnegative real numbers. A simple predictable process ξ is a left-continuous and adapted process which can be written as


for some n0, stopping times Sk<Tk, 0-measurable and bounded random variableMathworldPlanetmath A0 and Sk-measurable and bounded random variables Ak. In the case where Sk and Tk are deterministic times, then ξ is called an elementary predictable process.

The stochastic integral of the simple predictable process ξ with respect to a stochastic process Xt can then be written as

Title simple predictable process
Canonical name SimplePredictableProcess
Date of creation 2013-03-22 18:36:33
Last modified on 2013-03-22 18:36:33
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Definition
Classification msc 60G07
Defines simple predictable
Defines elementary predictable