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Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

Optimal portfolio selection with consumption and nonlinear integro-differential equations with... We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Lévy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Lévy process posseses only negative jumps. This is, however, the topic of a companion paper [7]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Finance and Stochastics Springer Journals

Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

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References (37)

Publisher
Springer Journals
Copyright
Copyright © 2001 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Quantitative Finance; Finance, general; Statistics for Business/Economics/Mathematical Finance/Insurance; Economic Theory/Quantitative Economics/Mathematical Methods; Probability Theory and Stochastic Processes
ISSN
0949-2984
DOI
10.1007/PL00013538
Publisher site
See Article on Publisher Site

Abstract

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Lévy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Lévy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

Journal

Finance and StochasticsSpringer Journals

Published: Jul 1, 2001

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