On the convergence of monotone schemes for path-dependent PDE

We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo [32] for viscosity solutions of path-dependent PDEs (PPDE), which extends the seminal work of Barles and Souganidis [1] on the viscosity solution of PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in [1]. These conditions are satisfied, to the best of our knowledge, by all classical monotone numerical schemes in the context of stochastic control theory. In particular, the paper provides a unified approach to prove the convergence of numerical schemes for non-Markovian stochastic control problems, second order BSDEs, stochastic differential games etc.