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:**

- 60G35 Applications (signal detection, filtering, etc.) [See also 62M20, 93E10, 93E11, 94Axx]
- 65C20 Models, numerical methods [See also 68U20]
- 65N50 Mesh generation and refinement
- 62L20 Stochastic approximation
- 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Résumé:** We present an approximation method for discrete time nonlinear
filtering in view of solving dynamic optimization problems under
partial information. The method is based on quantization of the
Markov pair process filter-observation $(\Pi,Y)$ and is such that,
at each time step $k$ and for a given size $N_k$ of the
quantization grid in period $k$, this grid is chosen to minimize a
suitable quantization error. The algorithm is based on a
stochastic gradient descent combined with Monte-Carlo simulations
of $(\Pi,Y)$. Convergence results are given and applications to
optimal stopping under partial observation are discussed.
Numerical results are presented for a particular stopping problem
: American option pricing with unobservable volatility.

**Mots Clés:** *Nonlinear filtering ; Markov chain ; quantization ; stochastic gradient descent ;
Monte-Carlo simulations ; partial observation ; optimal stopping
*

**Date:** 2004-09-07

**Prépublication numéro:** *PMA-925*