Feynman–Kac formula

Remarks

The proof applies Itô's formula to $Y_s = e^{-\int_t^s V(X_\tau, \tau) d\tau} u(X_s, s)$. By 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. The growth conditions and the moment bounds for $X$ make it a true martingale, after a localization by stopping times. Taking expectations between $s = t$ and $s = T$ gives the formula. Example: with $\mu = 0$, $\sigma = 1$, $V = f = 0$, the formula says that the solution of the backward heat equation $u_t + \frac{1}{2}u_{xx} = 0$ with $u(\cdot, T) = \psi$ is $u(x, t) = \mathbb{E}[\psi(x + W_{T} - W_t)]$, a Gaussian average of $\psi$. Without a growth condition on $u$ the conclusion can fail, because the heat equation has nonzero solutions with zero data that grow very fast in $x$. The 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 $\frac{1}{2}\sigma^2 \partial_x^2$.

Going deeper

For a student

The formula is the bridge between stochastic differential equations and second-order parabolic equations. The operator $\mu \partial_x + \frac{1}{2}\sigma^2 \partial_x^2$ is the infinitesimal generator of the diffusion $X$, and the potential $V$ acts as a killing rate. In finance it gives the risk-neutral pricing formula: the Black–Scholes equation is a special case with $V = r$ constant. The dual statement, for the density of $X$ rather than for expectations, is the Fokker–Planck equation. Its proof is the standard use of Itô's formula together with the optional stopping theorem.