A quantization algorithm for solving multidimensional discrete-time optimal stopping problems

A new grid method for computing the Snell envelope of a function of an Rᵈ-valued simulatable Markov chain (Xₖ)0λ ≤ kλ ≤ n is proposed. (This is a typical nonlinear problem that cannot be solved by the standard Monte Carlo method.) Every Xₖ is replaced by a `quantized approximation' Xₖ taking its values in a grid Γₖ of size Nₖ. The n grids and their trans\-ition probability matrices form a discrete tree on which a pseudo-Snell envelope is devised by mimicking the regular dynamic programming formula. Using the quantization theory of random vectors, we show the existence of a set of optimal grids, given the total number N of elementary Rᵈ-valued quantizers. A recursive stochastic gradient algorithm, based on simulations of (Xₖ)0λ ≤ k λ ≤ n, yields these optimal grids and their transition probability matrices. Some a priori error estimates based on the Lᵖ-quantization errors \|Xₖ- Xₖ\|ₚ are established. These results are applied to the computation of the Snell envelope of a diffusion approximated by its (Gaussian) Euler scheme. We apply these result to provide a discretization scheme for reflected backward stochastic differential equations. Finally, a numerical experiment is carried out on a two-dimensional American option pricing problem.