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  "body": "\n## Statement\n\nLet $T \u003e 0$ and let $(B_t)_{0 \\le t \\le T}$ be a standard [[Brownian motion|22735FGJV6]] on a probability space $(\\Omega, \\mathcal{F}, P)$.\nLet $(\\mathcal{F}_t)$ be the [[filtration|HNK1Y62G7V]] generated by $B$ and completed with the $P$-null sets.\nLet $X$ be an $\\mathcal{F}_T$-measurable random variable with $E[X^2] \u003c \\infty$.\nThen there is a progressively measurable process $H$ with $E\\left[\\int_0^T H_t^2 \\, dt\\right] \u003c \\infty$ such that\n$$X = E[X] + \\int_0^T H_t \\, dB_t,$$\nwhere the integral is the [[Itô integral|JA95NH206E]].\nThe process $H$ is unique up to a set of measure zero for $dt \\otimes dP$.\n\n## Remarks\n\nAs a consequence, every square-integrable [[martingale|0D0GBNDKDH]] $M$ for $(\\mathcal{F}_t)$ has the form $M_t = M_0 + \\int_0^t H_s \\, dB_s$, and so has a continuous version.\nFor the proof, the random variables $\\mathcal{E}\\left(\\int_0^{\\cdot} f \\, dB\\right)_T$ with deterministic $f$ are [[stochastic exponentials|3D88622PBQ]] with explicit representations.\nTheir linear span is dense in $L^2(\\mathcal{F}_T)$, and the [[Itô isometry|5MDK7HSYRM]] passes the representation to limits.\nExample: Itô's lemma gives $B_T^2 = T + \\int_0^T 2 B_t \\, dB_t$, so $H_t = 2 B_t$.\nThe filtration must be generated by $B$: if $\\mathcal{F}_0$ contains a nonconstant variable $\\xi$ independent of $B$ with $E[\\xi] = 0$, then $\\xi$ has no such representation, because $E\\left[\\int_0^T H \\, dB \\mid \\mathcal{F}_0\\right] = 0 \\neq \\xi$.\n\n## Explanations\n\n### L1 human\n\nIn a world where all randomness comes from one Brownian motion, every random payoff is a constant plus a stochastic integral against that motion.\nIn betting terms, any such payoff can be produced from its mean by a suitable trading strategy in the Brownian motion.\n\n### L2 human\n\nThe theorem is the mathematical basis of hedging in complete markets, such as the Black–Scholes model.\nCombined with the Girsanov theorem, it shows that each claim can be replicated by trading in the stock.\nThe Clark–Ocone formula of Malliavin calculus identifies the integrand $H$ for differentiable $X$.\nThe result also holds for a $d$-dimensional Brownian motion, with a vector-valued integrand.\n",
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