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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints


Code(s) de Classification MSC:

Résumé: This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The cla\-ssical dynamic programming approach leads to a characterization of the value function as viscosity solution of the highly nonlinear associated Bellman equation. By means of a logarithm transformation, we express the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist and expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.

Mots Clés: Stochastic volatility ; optimal portfolio ; dynamic programming equation ; logarithm transformation ; semilinear partial differential equation ; smooth solution

Date: 2001-03-27

Prépublication numéro: PMA-644