On the convergence of monotone schemes for path-dependent PDEs

We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo (2014) for viscosity solutions to path-dependent PDEs (PPDE), which extends the seminal work of Barles and Souganidis (1991) on the viscosity solution to PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in Barles and Souganidis (1991). 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.