## Bernoulli

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 23 December 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1072215199

**Digital Object Identifier**

doi:10.3150/bj/1072215199

**Mathematical Reviews number (MathSciNet)**

MR2046816

**Zentralblatt MATH identifier**

1042.60021

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.bj/1072215199