Title | A quantization algorithm for solving multidimensional optimal stopping problems, preprint |

Publication Type | Conference Paper |

Year of Publication | 2001 |

Authors | Vlad Bally, and Gilles Pagès |

Conference Name | University de Paris VI |

Abstract | A new grid method for computing the Snell envelope of a function of an -valued simulatable Markov chain is proposed. (This is a typical nonlinear problem that cannot be solved by the standard Monte Carlo method.) Every is replaced by a 'quantized approximation' taking its values in a grid of size . The grids and their transition 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 of elementary -valued quantizers. A recursive stochastic gradient algorithm, based on simulations of (, yields these 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 envelope of a diffusion approximated by its (Gaussian) Euler scheme. We apply these results 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. |