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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\ge 0}, P)$ be a filtered probability space with a [[filtration|HNK1Y62G7V]] that satisfies the usual conditions.\nLet $(B_t)_{t \\ge 0}$ be a [[Brownian motion|22735FGJV6]] for this filtration and let $T \u003e 0$.\n\n1. A *simple process* is a process $H_t = \\sum_{i=0}^{n-1} \\xi_i \\mathbf{1}_{(t_i, t_{i+1}]}(t)$, where $0 = t_0 \u003c t_1 \u003c \\cdots \u003c t_n = T$ and each $\\xi_i$ is a bounded $\\mathcal{F}_{t_i}$-measurable random variable.\nIts Itô integral is $\\int_0^T H_t \\, dB_t = \\sum_{i=0}^{n-1} \\xi_i (B_{t_{i+1}} - B_{t_i})$.\n2. Let $H$ be a progressively measurable process with $E\\left[\\int_0^T H_t^2 \\, dt\\right] \u003c \\infty$.\nThere are simple processes $H^{(n)}$ with $E\\left[\\int_0^T (H^{(n)}_t - H_t)^2 \\, dt\\right] \\to 0$.\nThe *Itô integral* $\\int_0^T H_t \\, dB_t$ is the limit in $L^2(P)$ of $\\int_0^T H^{(n)}_t \\, dB_t$.\nThe limit exists and does not depend on the choice of the $H^{(n)}$.\n\n*Itô calculus* is the study of these integrals, of their extensions to integrators that are semimartingales, and of the processes they define.\n\n## Remarks\n\nExample: for $H = \\mathbf{1}_{(0, T]}$, the integral is $B_T - B_0$.\nFor $H_t = B_t$, the left-point sums converge to $\\int_0^T B_t \\, dB_t = \\tfrac12 (B_T^2 - T)$ when $B_0 = 0$.\nNon-example: the process $H_t = B_T$ for all $t \u003c T$ is not [[adapted|YRNFE016FF]], so clause 2 does not apply to it.\nThe limit in clause 2 exists because of the [[Itô isometry|5MDK7HSYRM]], which makes the map $H \\mapsto \\int H \\, dB$ an isometry on simple processes.\nThe process $t \\mapsto \\int_0^t H_s \\, dB_s$ has a continuous version that is a square-integrable martingale.\nThe chain rule of Itô calculus is [[Itô's lemma|2J1RVQB0CM]].\n",
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