Stochastic integral

A stochastic integral is a generalization of the ordinary integral of calculus

$F = \int_I f(x)d\mu(x)$

where the integrator $\mu(x)$ is interpreted not as a deterministic function but as a stochastic process on the interval $I$. This means that, for each suitable function $f$, $F$ defined as above is a random variable. In this article we will restrict ourselves to the case where the integrator is Brownian motion (also known as the Wiener process) or, in physics terms, when the measure is white noise.

Integrals of deterministic functions with respect to Brownian motion $W(t)$ can be defined by the following property: if $f$ is of bounded variation and in $L^2[I]$, then $F = \int_I f(t)dW(t)$ is a Gaussian random variable with

$E[F] = 0\qquad and\quad Var[F] = \int_I f^2(t)dt.$

When the integrand is also stochastic, there are two main inequivalent definitions of the integral, the Ito Integral? and the Stratonovich integral.