Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

We design a numerical scheme for solving the Multi step-forward Dynamic Programming (MDP) equation arising from the time-discretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical least-squares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy. Numerical experiments are presented illustrating theoretical convergence estimates.