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  "body": "\n## Summary\n\nStochastic optimal control is [[optimal control|T3V6ZNAMDN]] of systems driven by noise, usually modelled by a [[stochastic differential equation|B219G9SHJA]].\nThe controller chooses its action at each time from the information observed so far and minimises an expected cost.\nDynamic programming leads to a second-order [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] for the value function.\nThe theory is used in engineering, in economics and in mathematical finance, for example in portfolio choice.\n\n## Explanations\n\n### L1 human\n\nThink of steering a boat in gusty wind.\nYou cannot plan the whole path in advance, because the wind keeps pushing you off course.\nInstead you decide at each moment what to do given where you are now.\nA simple model is a position $X_t$ that moves with the chosen velocity $u_t$ plus a random [[Brownian|22735FGJV6]] kick.\nThe goal is to end near $0$ without spending too much effort.\nThe best rule turns out to be a feedback: push back towards $0$ in proportion to the current position.\n",
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