Itô integral

This is the technical formulation that makes it meaningful to write stochastic differential equations 'driven by' a Weiner process $W$, such as

The construction is nontrivial because the Wiener process is almost surely nowhere differentiable, so the differential $dW$ is not defined in the conventional sense. The Itô integral is essentially a procedure for evaluating integrals with respect to $dW$ and other stochastic processes, which (following the fundamental theorem of calculus) defines the corresponding differentials. Essentially, we

The full construction is built up from the basic case of integrating a constant function $\beta(x, t; \omega) = \bar{\beta}$ over a range: for any sample path $W(\omega)$ we define

This can be generalized by appropriate constructions to define integrals $I[\beta](\omega)$ of piecewise constant functions $\beta(t)$, random piecewise constant functions $\beta(t; \omega)$, and arbitrary random functions $\beta(t; \omega)$ built up as limits of piecewise constant functions. Note that the integrand $\beta$ must be adapted to the same filtration as $W_t$; i.e., it can't depend on future information, only what would be available at time $t$.