Solving a Backward SDE with the Cubature method

Probabilistic methods for the solution of Backward Stochastic Differential Equations (BSDE) provide us with a new approach to the problem of approximating the solution of a semi-linear PDE. Utilizing on the Markovian nature of these BSDE’s we show how one may consider the problem of numerical solutions to BSDEs within the area of weak approximations of diffusions. To emphasize this point, we suggest an algorithm based on the Cubature method on Wiener space of Lyons - Victoir. Instead of using standard discretization techniques of BSDE’s, we choose to work with the actual flow. This allows to take advantage of estimates on the derivatives of the solution of the associated semi-linear PDE and hence, we recover satisfactory convergence estimates.