Let $X_t$ be the $n$ -dimensional Ito process satisfying the stochastic differential equation $$ dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t $$ and let $A$ be its infinitesimal generator.

Further suppose that $q$ is a lower-bounded continuous function on $\real^n$ , and $f$ is a twice-differentiable function on $\real^n$ with compact support. Then $$ u(t,x) = \E\bigl[ e^{-\int_0^t q(X_s) \, ds} \, f(X_t) \mid X_0 = x \bigr]\,, \quad t\geq 0\,, x \in \real^n $$ is a solution to the partial differential equation $$ \pd{u}{t} = Au(x) - uq(x) $$ with initial condition $u(0, x) = f(x)$ .

(The expectation for $u$ is to be taken with respect to the probability measure under which $W_t$ is a Brownian motion.)

Bibliography

1

Bernt Øksendal. Stochastic Differential Equations, An Introduction with Applications. 5th ed., Springer 1998.

2

Hui-Hsiung Kuo. Introduction to Stochastic Integration. Springer 2006.