Connections between SDE and PDE

Question

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include:

1. The heat equation $u_t = \frac{1}{2} \Delta u$ with initial data $u(0,x)=f(x)$, considered on the whole space. Here the solution is given by $u(t,x)=\mathbb{E}(f(x+W_t))$ where $W_t$ is a Wiener process.

2. Suppose $U$ is a bounded open set, $X_t^x$ is a Markov process which starts at $x$ and has generator $L$, and $\tau_x = \inf \{ t \geq 0 : X_t^x \in \partial U \}$. Then $u(x) = \mathbb{E}(\tau_x)$ solves

$$(Lu)(y) = -1 \text{ if } y \in U \\ u(y) = 0 \text{ if } y \in \partial U$$

for $x \in \overline{U}$. When $X_t^x$ is a diffusion process this is an elliptic or semielliptic boundary value problem.

The most general formula for such situations that I have seen is the Feynman-Kac formula, though I have seen hints that the Feynman-Kac formula is itself a special case of Girsanov's formula, which I have some difficulty understanding. I am looking for a relatively comprehensive discussion of this phenomenon, preferably with some applications or at least additional toy examples.

Answer 1

For future reference, I found a nice source myself:

http://www3.ntu.edu.sg/home/nprivault/papers/greifswald_potential.pdf

Sections 1 and 2 are on potential theory. Sections 3 and 4 are on probability. Section 5 presents a number of the sorts of connections I was looking for here. A bit of overlap is splashed between the first four sections. It is a very nice source although it would have been better for my purposes if Section 5 had a bit more content.

Answer 2

This is a quite classical topic, subject of several graduate courses in a lot of universities. A bunch of applications are in Monte Carlo simulation is solutions of PDE in high dimension (where the finite differences methods become inefficient), in particular in Finance.

For advanced extensions to finance, I can recommend you a book from Pierre Henry Labordere: Nonlinear Option Pricing.