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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

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

- 65-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
- 90C39 Dynamic programming [See also 49L20]
- 93E35 Stochastic learning and adaptive control
- 91B28 Finance, portfolios, investment
- 65L08 Improperly posed problems
- 60G35 Applications (signal detection, filtering, etc.) [See also 62M20, 93E10, 93E11, 94Axx]
- 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Résumé:** We review optimal quantization methods for numerically solving
nonlinear problems in higher dimension associated with Markov
processes. Quantization of a Markov process consists
in a spatial discretization on finite grids optimally fitted
to the dynamics of the process.
Two quantization methods are proposed: the first one, called
marginal quantization, relies on an optimal
approximation of the marginal distributions of the process,
while the second one, called Markovian quantization, looks for an
optimal approximation of transition
probabilities of the Markov process at some points. Optimal grids and
their associated weights
can be computed by a stochastic gradient descent method based on
Monte Carlo simulations. We illustrate this optimal quantization
approach with four numerical applications arising in finance:
European option pricing, optimal stopping problems and American
option pricing, stochastic
control problems and mean-variance hedging of options and filtering
in stochastic volatility models.

**Mots Clés:** *Quantization ; Markov chain ; Euler scheme ; Numerical integration ; Optimal stopping ; Option pricing ; Stochastic control ;
Non linear filtering ; Stochastic gradient descent*

**Date:** 2003-04-24

**Prépublication numéro:** *PMA-813*

**Front pages :** PMA-813.dvi