On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities

The classical Feynman-Kac identity represents solutions of linear partial differential equations in terms of stochastic differential euqations. This representation has been generalized to nonlinear partial differential equations on the one hand via backward stochastic differential equations and on the other hand via stochastic fixed-point equations. In this article we generalize the representation via stochastic fixed-point equations to allow the nonlinearity in the semilinear partial differential equation to depend also on the gradient of the solution.