Itô calculus

Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a filtered probability space with a filtration that satisfies the usual conditions. Let $(B_t)_{t \ge 0}$ be a Brownian motion for this filtration and let $T > 0$.

1. A simple process is a process $H_t = \sum_{i=0}^{n-1} \xi_i \mathbf{1}_{(t_i, t_{i+1}]}(t)$, where $0 = t_0 < t_1 < \cdots < t_n = T$ and each $\xi_i$ is a bounded $\mathcal{F}_{t_i}$-measurable random variable. Its Itô integral is $\int_0^T H_t \, dB_t = \sum_{i=0}^{n-1} \xi_i (B_{t_{i+1}} - B_{t_i})$.
2. Let $H$ be a progressively measurable process with $E\left[\int_0^T H_t^2 \, dt\right] < \infty$. There are simple processes $H^{(n)}$ with $E\left[\int_0^T (H^{(n)}_t - H_t)^2 \, dt\right] \to 0$. The Itô integral $\int_0^T H_t \, dB_t$ is the limit in $L^2(P)$ of $\int_0^T H^{(n)}_t \, dB_t$. The limit exists and does not depend on the choice of the $H^{(n)}$.

Itô calculus is the study of these integrals, of their extensions to integrators that are semimartingales, and of the processes they define.