ABSTRACT
In Chapter 7, we introduced first and second order (Markovian) backward stochastic differential equations (BSDEs). They provide a generalization of the Feynman-Kac theorem for nonlinear PDEs. Unfortunately, in practice, numerically solving BSDEs requires the computation of conditional expectations, typically using regression methods. Finding regressors of good quality is difficult, notably for multi-asset portfolios. This leads us to introduce a new promising method based on branching diffusions describing a marked Galton-Watson random tree. We first show how these branching diffusions can provide stochastic representations for solutions of a large class of semilinear parabolic PDEs in which the nonlinearity can be approximated by a polynomial function. We then briefly extend our method to fully nonlinear PDEs. Numerical examples, including the computation of the counterparty risk, illustrates the efficiency of our algorithm. Parts of this research have been published in [130] and [132], and presented in [131].
