= 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 $\mathbb E(\int_0^tH_s\,dB_s)^2=\mathbb E\int_0^tH_s^2\,ds$ for square-integrable adapted integrands. The resulting <Itô formula> includes a second-derivative correction:
$$
d(B_t^2)=2B_t\,dB_t+dt,\qquad B_t^2=2\int_0^tB_s\,dB_s+t.
$$
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.<Applied Stochastic Differential Equations by Simo Särkkä and Arno Solin>[Chapter 4, Itô calculus, stochastic integration and Itô formula]{ref}
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