Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 65C20 Models, numerical methods
- 65N50 Mesh generation and refinement
- 90C39 Dynamic programming, See also {49L20}
- 93E35 Stochastic learning and adaptive control
- 90A09 Finance, portfolios, investment

**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.

**Mots Clés:** *Stochastic control ; Markov chain ; Euler scheme ; Vector quantization ;
Stochastic gradient descent*

**Date:** 2001-11-07

**Prépublication numéro:** *PMA-697*