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  "body": "\n## Summary\n\nA stochastic differential equation describes a quantity whose change in a short time is a drift plus a random kick.\nThe kick is proportional to the increment of a [[Wiener process|22735FGJV6]].\nIt is written $dX_t = b(t, X_t) \\, dt + \\sigma(t, X_t) \\, dW_t$ and is made precise as an integral equation with an Itô integral from [[Itô calculus|JA95NH206E]].\nSuch equations model stock prices, noisy physical systems, and population sizes under random effects, and their solutions are diffusion processes.\n\n## Explanations\n\n### L1 human\n\nAn ordinary differential equation tells a particle exactly where to go next.\nA stochastic differential equation adds a random push at each moment, built from the jittery path of Brownian motion.\nThe drift $b$ says where the particle tends to move, and the coefficient $\\sigma$ says how strong the noise is.\nFor example, the solution of $dX_t = X_t \\, dW_t$ with $X_0 = 1$ is $X_t = e^{W_t - t/2}$, not $e^{W_t}$, because the rules of calculus change for such rough paths.\n",
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