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:**

- 60H30 Applications of stochastic analysis (to PDE, etc.)
- 90A09 Finance, portfolios, investment
- 93E20 Optimal stochastic control
- 35K55 Nonlinear PDE of parabolic type

**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*