Ito Integral for Simple Processes

Simple adapted process

Suppose

$$ 0=t_0and

$$ Y_t=Y^{(k)} \qquad \text{for } t\in [t_k,t_{k+1}), $$

where each $Y^{(k)}$ is $\mathcal F_{t_k}$-measurable.

For such a process, define the Ito integral by

\sum_{k=0}^{m-1} Y^{(k)}(B_{t_{k+1}}-B_{t_k}). $$

On each interval, the integrand is frozen at the left endpoint and multiplied by the Brownian increment on that interval.