A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders

Abstract

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations with arbitrary gradient nonlinearities. This algorithm extends the classical Feynman-Kac formula to fully nonlinear PDEs using random trees that carry information on nonlinearities on their branches. It applies to functional, nonpolynomial nonlinearities that cannot be treated by standard branching arguments and deals with gradients of any orders, avoiding the integrability issues encountered with the use of Malliavin-type weights.

Time-inconsistent control problems, path-dependent PDEs, and neural network approximation

“…[MHJKN20]. Therefore, using a similar idea, in [NPP22] we have extended the stochastic branching method of [PP22] to obtain full functional solution estimates of high-dimensional fully nonlinear PDEs. The future work can focus on extending this method to solve PDE systems such as the Navier-Stokes equation and the equilibrium HJB system that corresponds to the TIC problem, see e.g.…”