Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A quantization algorithm for multidimensional stochastic control problems

Auteur(s):

Code(s) de Classification MSC:

• 65C20 Models, numerical methods
• 65N50 Mesh generation and refinement
Résumé: We propose a probabilistic numerical method based on quantization to solve some multidimensional stochastic control problems that arise, $e.g.$, in Mathematical Finance for portfolio optimization purpose. This leads to consider some controlled diffusions with most control free components. The space discretization of this partof the diffusion is achieved by a closest neighbour projection of the Euler scheme increments of the diffusion on some grids. The resulting process is a discrete time inhomogeneous Markov chain with finite state spaces. The induced control problem can be solved using the dynamic programming formula. {\em A priori} $L^p$-error bounds are produced and we show that the space discretization error term is minimal at some specific grids. A simple recursive algorithm is devised to compute these grids by induction based on a Monte Carlo simulation.