Convergence of particles and tree based scheme for singular FBSDEs

We study an implementation of the theoretical splitting scheme introduced in [Chassagneux and Yang, 2022] for singular FBSDEs [Carmona and Delarue 2013] and their associated quasi-linear degenerate PDEs. The fully implementable algorithm is based on particles approximation of the transport operator and tree like approximation of the diffusion operator appearing in the theoretical splitting. We prove the convergence with a rate of our numerical method under some reasonable conditions on the coefficients functions. This validates a posteriori some numerical results obtained in [Chassagneux and Yang, 2022]. We conclude the paper with a numerical section presenting various implementations of the algorithm and discussing their efficiency in practice.