Definition Simple Stochastic Process

Question

Can anybody explain what concretely is a simple (stochastic) process? (...and yes I tried to google it but without success)

Answer 1

A simple stochastic process $f$ on $[0,T]$ is one for which there exists times $0=t_1<t_1<t_2<\dots<t_n=T$ such that $f(t)$ is almost surely constant (in $t$) on each interval $[t_{i-1},t_i)$. This means that there are random variables $\xi_i,\dots,\xi_n$ such that $$ f(t) = \sum_{i=1}^n\xi_i\cdot {\bf 1}_{[t_{i-1},t_i)}(t) $$ Additionally, if there is a filtration $(\mathcal F_t)_{t\in [0,T]}$, then it is required that $f$ be adapted to this filtration, which is equivalent to saying that $\xi_i\in \mathcal F_{t_{i-1}}$.

Just like simple functions are used to define the integral in measure theory, simple processes are used to define the stochastic integral $\int f(t)\,dB_t$.