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  "body": "\n## Statement\n\nLet $T \u003e 0$ and let $W$ be a one-dimensional [[Wiener process|22735FGJV6]].\nLet $\\mu, \\sigma \\colon \\R \\times [0, T] \\to \\R$ be continuous and Lipschitz in $x$ with linear growth, uniformly in $t$.\nFor $(x, t) \\in \\R \\times [0, T)$, let $X^{x,t}$ be the solution of the [[stochastic differential equation|B219G9SHJA]]\n$$dX_s = \\mu(X_s, s)\\, ds + \\sigma(X_s, s)\\, dW_s, \\quad s \\ge t, \\qquad X_t = x.$$\nLet $V \\colon \\R \\times [0, T] \\to [0, \\infty)$ be continuous, and let $f \\colon \\R \\times [0, T] \\to \\R$ and $\\psi \\colon \\R \\to \\R$ be continuous with polynomial growth in $x$.\nLet $u \\colon \\R \\times [0, T] \\to \\R$ be continuous, of class $C^{2,1}$ on $\\R \\times [0, T)$, of polynomial growth in $x$ uniformly in $t$, and a solution of the [[parabolic problem|1CCTH8X2FT]]\n$$u_t + \\mu\\, u_x + \\frac{1}{2}\\sigma^2 u_{xx} - V u + f = 0 \\ \\text{ on } \\R \\times [0, T), \\qquad u(x, T) = \\psi(x).$$\nThen for all $(x, t) \\in \\R \\times [0, T]$,\n$$u(x, t) = \\mathbb{E}\\Big[ \\int_t^T e^{-\\int_t^r V(X_\\tau, \\tau)\\, d\\tau} f(X_r, r)\\, dr + e^{-\\int_t^T V(X_\\tau, \\tau)\\, d\\tau}\\, \\psi(X_T) \\Big],$$\nwhere $X = X^{x,t}$.\n\n## Remarks\n\nThe proof applies [[Itô's formula|2J1RVQB0CM]] to $Y_s = e^{-\\int_t^s V(X_\\tau, \\tau) d\\tau} u(X_s, s)$.\nBy the equation for $u$, the drift of $Y_s$ equals $-e^{-\\int_t^s V d\\tau} f(X_s, s)$, so $Y_s$ plus the integral of this term is a local martingale.\nThe growth conditions and the moment bounds for $X$ make it a true martingale, after a localization by stopping times.\nTaking expectations between $s = t$ and $s = T$ gives the formula.\nExample: with $\\mu = 0$, $\\sigma = 1$, $V = f = 0$, the formula says that the solution of the backward [[heat equation|VP2HN86VAG]] $u_t + \\frac{1}{2}u_{xx} = 0$ with $u(\\cdot, T) = \\psi$ is $u(x, t) = \\mathbb{E}[\\psi(x + W_{T} - W_t)]$, a Gaussian average of $\\psi$.\nWithout a growth condition on $u$ the conclusion can fail, because the heat equation has nonzero solutions with zero data that grow very fast in $x$.\nThe same formula holds in $\\R^n$ with $\\frac{1}{2}\\sum_{i,j} (\\sigma\\sigma^T)_{ij}\\, \\partial_{x_i}\\partial_{x_j}$ in place of $\\frac{1}{2}\\sigma^2 \\partial_x^2$.\n\n## Explanations\n\n### L1 human\n\nThe Feynman–Kac formula says that the solution of a certain parabolic equation is an average over random paths.\nStart a random walker at the point $x$ at time $t$, let it wander until time $T$, and record a reward collected along the way and at the end.\nThe average reward is the value $u(x, t)$.\nThis lets one compute solutions of equations by simulating random paths, and it lets one study random paths by solving equations.\n\n### L2 human\n\nThe formula is the bridge between [[stochastic differential equations|B219G9SHJA]] and second-order parabolic equations.\nThe operator $\\mu \\partial_x + \\frac{1}{2}\\sigma^2 \\partial_x^2$ is the [[infinitesimal generator|VPA41PWRR1]] of the diffusion $X$, and the potential $V$ acts as a killing rate.\nIn finance it gives the risk-neutral pricing formula: the Black–Scholes equation is a special case with $V = r$ constant.\nThe dual statement, for the density of $X$ rather than for expectations, is the Fokker–Planck equation.\nIts proof is the standard use of Itô's formula together with the optional stopping theorem.\n",
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