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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_t)_{t \\ge 0}$.\nLet $(X_t)_{t \\ge 0}$ be a real [[adapted|YRNFE016FF]] process with continuous paths and $X_0 = 0$.\nAssume that $(X_t)$ and $(X_t^2 - t)$ are both [[local martingales|ZWJ9G89W3T]].\nThen $(X_t)$ is a [[Wiener process|552AWX3QXN]], and for all $0 \\le s \u003c t$ the increment $X_t - X_s$ is independent of $\\mathcal{F}_s$.\n\n## Remarks\n\nThe second hypothesis says that the [[quadratic variation|ZBYC3MFW89]] of $X$ is $[X]_t = t$.\nFor a fixed $u \\in \\R$, Itô's formula shows that $Z_t = \\exp(i u X_t + u^2 t / 2)$ is a local martingale; it is bounded on bounded time intervals, hence a martingale.\nSo $E[\\exp(i u (X_t - X_s)) \\mid \\mathcal{F}_s] = \\exp(-u^2 (t - s)/2)$, which identifies the conditional law of the increment as normal with mean $0$ and variance $t - s$.\nContinuity is necessary: for a Poisson process $N$ of rate $1$, both $N_t - t$ and $(N_t - t)^2 - t$ are martingales, but $N_t - t$ is not a Wiener process.\nIn dimension $d$, a continuous process $X$ with $X_0 = 0$ is a $d$-dimensional Wiener process if each $X^i$ and each $X^i X^j - \\delta_{ij} t$ is a local martingale.\n\n## Explanations\n\n### L1 human\n\nBrownian motion is the only continuous fair game whose squared size grows on average exactly like time.\nNothing about normal distributions or independence has to be assumed: they come for free.\nSo these two martingale properties pin down Brownian motion completely.\n\n### L2 human\n\nThe theorem is the standard way to recognise a Brownian motion inside stochastic calculus.\nIt gives the Dambis–Dubins–Schwarz theorem: a continuous local martingale $M$ with $[M]_\\infty = \\infty$ is a time-changed Brownian motion, $M_t = B_{[M]_t}$.\nIt is also a step in the proof of the [[Girsanov theorem|Z76HD1BZ29]], and it shows that $\\int_0^t \\sgn(W_s) \\, dW_s$ is a Wiener process, which is used with [[Tanaka's formula|GEM0ERJ5ZB]].\n",
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