Conditional Monte Carlo Learning for Diffusions I: main methodology and application to backward stochastic differential equations.

Summary We present a new algorithm based on a One-layered Nested Monte Carlo (1NMC) to simulate functionals U of a Markov process X. The main originality of the proposed methodology comes from the fact that it provides a recipe to simulate U_{t≥s} conditionally on X_s. Because of the nested structure that allows a Taylor-like expansion, it is possible to use a very reduced basis for the regression. Although this methodology can be adapted for a large number of situations, we only apply it here for the simulation of Backward Stochastic Differential Equations (BSDEs). The generality and the stability of this algorithm, even in high dimension, make its strength. It is heavier than a straight Monte Carlo (MC) but it is far more accurate to simulate quantities that are almost impossible to simulate with MC. The parallel suitability of 1NMC makes it feasible in a reasonable computing time. This paper explains the main version of this algorithm and provides first results of error estimates. We also give various numerical examples with a dimension equal to 100 that are executed from few seconds to few minutes on one Graphics Processing Unit (GPU).