A Generalized Feynman-Kac Formula For One Dimensional Processes

Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f , the Feynman-Kac formula gives a condition for f(t,X) to be a local martingale. We generalize the Feynman-Kac formula in two main ways. First, it is extended to nondifferentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability condition, and the martingale condition for f(t,X) is then expressed in terms of the marginal distributions, drift measure and jumps of X . The proof involves the stochastic calculus of Dirichlet processes and a time-reversal argument. These results are then applied to show that a continuous and strong Markov martingale is uniquely determined by its marginal distributions.