• Bernoulli
  • Volume 9, Number 6 (2003), 1003-1049.

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

Vlad Bally and Gilles Pagès

Full-text: Open access


A new grid method for computing the Snell envelope of a function of an $\mathbb{R}^d$-valued simulatable Markov chain $(X_k)_{0\lambda \leq k\lambda \leq n}$ is proposed. (This is a typical nonlinear problem that cannot be solved by the standard Monte Carlo method.) Every $X_k$ is replaced by a `quantized approximation' $\widehat{X}_k$ taking its values in a grid $\Gamma_k$ of size $N_k$. 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 $\mathbb{R}^d$-valued quantizers. A recursive stochastic gradient algorithm, based on simulations of $(X_k)_{0\lambda \leq k \lambda \leq n}$, yields these optimal grids and their transition probability matrices. Some a priori error estimates based on the $L^p$-quantization errors $\|X_k-\widehat X_k\|_{_p}$ 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.

Article information

Bernoulli, Volume 9, Number 6 (2003), 1003-1049.

First available in Project Euclid: 23 December 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

American option pricing Markov chains numerical probability quantization of random variables reflected backward stochastic differential equation Snell envelope


Bally, Vlad; Pagès, Gilles. A quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli 9 (2003), no. 6, 1003--1049. doi:10.3150/bj/1072215199.

Export citation