Probabilistic numerical methods for fully nonlinear PDEs and related topics

In this research we will discuss three topics, starting with a monotone scheme for high dimensional fully nonlinear PDEs, which include the quasi-linear PDE that corresponds to a coupled FBSDE as a special case. This work is strongly motivated by the remarkable work by Fahim, Touzi and Warin, and stays in the paradigm of monotone schemes initiated by Barles and Souganidis . It weakens a critical constraint imposed by, especially when the generator of the PDE depends only on the diagonal terms of the hessian matrix. Several numerical examples, up to dimension 12, are reported. The second part is dedicated to monotone schemes for fully nonlinear pathdependent PDEs. This is an extension of the seminal work Barles and Souganidis to path dependent case. Based on the viscosity theory of path dependent PDEs, developed by Ekren, Keller, Touzi and Zhang and Ekren, Touzi and Zhang , we show that a monotone scheme converges to the unique viscosity solution of the (fully nonlinear) parabolic path dependent PDE.