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  "body": "\n## Statement\n\nLet $(B_t)_{t \\ge 0}$ be a [[Brownian motion|22735FGJV6]] for a filtration $(\\mathcal{F}_t)$, and let $T \u003e 0$.\nLet $H$ be a progressively measurable process with $E\\left[\\int_0^T H_t^2 \\, dt\\right] \u003c \\infty$.\nThen the [[Itô integral|JA95NH206E]] satisfies\n$$E\\left[\\left(\\int_0^T H_t \\, dB_t\\right)^2\\right] = E\\left[\\int_0^T H_t^2 \\, dt\\right],$$\nwhere $E$ is the [[expected value|0DCT0QZVQ1]].\n\n## Remarks\n\nFor a simple process $H = \\sum_i \\xi_i \\mathbf{1}_{(t_i, t_{i+1}]}$, expand the square of $\\sum_i \\xi_i \\Delta B_i$.\nA cross term $E[\\xi_i \\xi_j \\Delta B_i \\Delta B_j]$ with $i \u003c j$ vanishes, because $\\Delta B_j$ has mean zero and is independent of $\\mathcal{F}_{t_j}$.\nA diagonal term equals $E[\\xi_i^2] (t_{i+1} - t_i)$, because $\\Delta B_i$ is independent of $\\mathcal{F}_{t_i}$ with variance $t_{i+1} - t_i$.\nThe general case follows by approximation, and this is how the integral is defined.\nExample: for $H_t = B_t$ with $B_0 = 0$, the right side is $\\int_0^T t \\, dt = T^2/2$, and indeed $\\int_0^T B_t \\, dB_t = \\tfrac12 (B_T^2 - T)$ has variance $\\tfrac14 (3T^2 - 2T^2 + T^2) = T^2/2$.\n\n## Explanations\n\n### L1 human\n\nThe Itô integral of $H$ is a random number with mean zero.\nIts variance equals the expected total of $H^2$ over time.\nSo the size of the integral can be computed by an ordinary integral, without the random noise.\n\n### L2 human\n\nThe isometry is the key step in the construction of the Itô integral: it extends the integral from simple processes to all square-integrable adapted ones.\nIts polarised form gives $E\\left[\\int H \\, dB \\int K \\, dB\\right] = E\\left[\\int H K \\, dt\\right]$.\nFor a general continuous martingale $M$ in place of $B$, the same identity holds with $dt$ replaced by $d\\langle M \\rangle_t$.\n",
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  "name": "Itô isometry",
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