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) $({ℱ}_t)$ on the measurable space $(\mathrm{\Omega },ℱ)$ 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

for some $n\ge 0$, stopping times , ${ℱ}_0$-measurable and bounded random variable $A_0$ and ${ℱ}_{S_k}$-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

$${\int }_0^t\xi 𝑑X=\sum _{k=1}^n{1}_{\{t>S_k\}}{A}_k(X_{\min (t,T_k)}-X_{S_k}).$$