Diffusions, SDEs, and the Feynman-Kac Formula — Overview

Itô's formula takes a process and tells you how smooth functions of it evolve. The natural next question is the inverse: given a desired dynamics $dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dW_t$, does such a process exist? The answer — yes, under Lipschitz conditions, by Picard iteration — is the existence-uniqueness theorem for SDEs, and the process is called a diffusion.

The payoff of solving an SDE is the Feynman-Kac formula: a deep connection between diffusions and PDEs. Specifically, the solution $u(t, x) = \mathbb{E}[g(X_T) e^{-\int_t^T c(X_s)ds} | X_t = x]$ to a terminal-value problem satisfies a parabolic PDE — and conversely, the solution to the PDE can be represented as such an expectation. In finance, this is the link between option pricing (an expectation under $\mathbb{Q}$) and the Black-Scholes PDE (a parabolic equation for the option price). These are not two separate methods; they are two faces of the same object.