A Nonlinear Feynman-Kay Formula with Application in Linearly Solvable Optimal Control

In this article we present a solution to a nonlinear relative of the parabolic differential equation that was tackled by Feynman and Kac in the late 1940s. For the proof we rely on continuous time stochastic calculus. Second we draw an interesting connection with a related recurrence relation affirming the presented result by collapsing onto the continuous time framework but only in the limit. The equation emerges in the context of infinite horizon discounted Linearly Solvable Optimal Control, which, as far as we are aware of, is untreated by the literature. The continuous time setting can be treated using our new result. As we will demonstrate the discrete time setting is intractable. Nevertheless we can provide close estimates based on the recurrence relation which also allows us to estimate the influence of time discretization errors. We demonstrate our solution treating a small case study.