 Stochastic analysis

Stochastic analysis develops calculus and analytic tools for stochastic processes, especially when their sample paths are too irregular for ordinary differentiation. Brownian motion is the central example: its increments have size of order $\sqrt{\Delta t}$ rather than $\Delta t$, and its paths are almost surely nowhere differentiable.

The Itô integral is defined through an appropriate limit of sums whose integrands use information available before each increment. Its Itô isometry gives $\mathbb{E}({\int }_0^t{H}_{s}d{B}_s{)}^2=\mathbb{E}{\int }_0^t{H}_s^{2}ds$ for square-integrable adapted integrands. The resulting Itô formula includes a second-derivative correction:Taking expectations recovers $\mathbb{E}{B}_t^2=t$. Ordinary calculus would miss the extra term and predict the wrong variance. This is why stochastic calculus is needed for stochastic differential equations and their stochastic partial differential equation counterparts, rather than treating random trajectories as differentiable curves.^[ref]