simple predictable process

Simple predictable processes are a particularly simple class of stochastic processes, 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 filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) $(\mathcal{F}_{t})$ on the measurable space $(\Omega,\mathcal{F})$ with time index $t$ ranging over the nonnegative real numbers. A simple predictable process $\xi$ is a left-continuous and adapted process which can be written as

\xi_{t}=1_{\{t=0\}}A_{0}+\sum_{k=1}^{n}1_{\{S_{k}<t\leq T_{k}\}}A_{k}

for some $n\geq 0$ , stopping times $S_{k}<T_{k}$ , $\mathcal{F}_{0}$ -measurable and bounded random variable $A_{0}$ and $\mathcal{F}_{S_{k}}$ -measurable and bounded random variables $A_{k}$ . In the case where $S_{k}$ and $T_{k}$ are deterministic times, then $\xi$ is called an elementary predictable process.

The stochastic integral of the simple predictable process $\xi$ with respect to a stochastic process $X_{t}$ can then be written as

\int_{0}^{t}\xi\,dX=\sum_{k=1}^{n}1_{\{t>S_{k}\}}A_{k}(X_{\min(t,T_{k})}-X_{S_% {k}}).

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