• G. PAGÈS
• H. PHAM
• J. PRINTEMS

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