Dynkin's formula

Dynkin's formula is the practical bridge from a Markov process to its generator: it lets you compute the expected value of an observable at a (random) stopping time by integrating the generator along the path. It is the optional-stopping theorem applied to the Dynkin martingale, and it is the everyday tool for hitting probabilities, mean exit times and harmonic-function arguments.

Start from the central martingale identity: for f in the domain D(A), the process M_t^f = f(X_t) - f(X_0) - integral_0^t (A f)(X_s) ds is a martingale (this is precisely the martingale-problem characterization). Taking expectations gives E_x[ f(X_t) ] - f(x) = E_x[ integral_0^t A f(X_s) ds ]. Dynkin's formula is the version at a stopping time tau: provided tau has finite expectation (or M^f is suitably uniformly integrable up to tau), optional stopping yields E_x[ f(X_tau) ] = f(x) + E_x[ integral_0^tau (A f)(X_s) ds ]. In words: the expected change in f from start to the stopping time equals the expected accumulated generator-rate of f over the elapsed time.

Why it matters: choosing f cleverly turns this into closed-form answers. If A f = 0 (f is A-harmonic) then E_x[ f(X_tau) ] = f(x), recovering hitting probabilities; if A f = -1 then E_x[ f(X_tau) ] = f(x) - E_x[tau], so solving A u = -1 with the right boundary data gives mean exit times; the connection to the Dirichlet problem and Feynman-Kac is the same idea with a potential term. The integrability hypothesis on tau is not optional: for unbounded domains or heavy-tailed exit times the formula can fail (the martingale is only local, and naive optional stopping is illegitimate) — exactly the same strict-local-martingale caution that haunts continuous-time martingale theory.