A quantization algorithm for solving multi-dimensional Optimal Stopping problems

A new grid method for computing the Snell envelop of a function of a R d -valued Markov chain (Xk)0≤k≤n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every Xk is replaced by a “quantized approximation” X̂k taking its values in a grid Γk of size Nk. The n grids and their transition probability matrices make up a discrete tree on which a pseudo-Snell envelop is devised by mimicking the regular dynamic programming formula. We show, using Quantization Theory of probability distributions the existence of a set of optimal grids, given the total number N of elementary R d -valued vector quantizers. A recursive stochastic algorithm, based on some simulations of (Xk)0≤k≤n, yields the optimal grids and their transition probability matrices. Some a priori error estimates based on the quantization errors are established. These results are applied to the computation of the Snell envelop of a diffusion (assuming that it can be directly simulated or using its Euler scheme). We show how this approach yields a discretization method for Reflected Backward Stochastic Differential Equation. Finally, some first numerical tests are carried out on a 2-dimensional American option pricing problem.