Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints |
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Authors: | Pham |
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Institution: | (1) Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, UFR Mathématiques, Case 7012, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France pham@gauss.math.jussieu.fr and CREST, Laboratoire de Finance-Assurance, 92 245 Malakoff Cedex, France, FR |
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Abstract: |
Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility
and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as
a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses 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 the existence of a smooth solution to this semilinear
equation. An optimal portfolio is shown to exist, and is 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. |
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Keywords: | , Stochastic volatility, Optimal portfolio, Dynamic programming equation, Logarithm transformation, Semilinear partial,,,,,differential equation, Smooth solution, AMS Classification, 60H30, 90A09, 93E20, 35K55, |
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