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Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution

Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal... We consider an optimal portfolio-consumption problem which incorporates the notions of durability and intertemporal substitution. The logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve Lévy processes as driving noise instead of the more frequently used Brownian motion. The optimization problem is a singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a nonlinear second order degenerate elliptic integro-differential equation subject to gradient and state constraints. For utility functions of HARA type, we calculate the optimal investment and consumption policies together with an explicit expression for the value function when the Lévy process has only negative jumps. For the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies for general Lévy processes having both positive and negative jumps. Instead of following the classical approach of using a verification theorem, we validate our solution candidates within a viscosity solution framework. To this end, the value function of our singular control problem is characterized as the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation in the case of general utilities and general Lévy processes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Finance and Stochastics Springer Journals

Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution

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References (17)

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/s007800000032
Publisher site
See Article on Publisher Site

Abstract

We consider an optimal portfolio-consumption problem which incorporates the notions of durability and intertemporal substitution. The logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve Lévy processes as driving noise instead of the more frequently used Brownian motion. The optimization problem is a singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a nonlinear second order degenerate elliptic integro-differential equation subject to gradient and state constraints. For utility functions of HARA type, we calculate the optimal investment and consumption policies together with an explicit expression for the value function when the Lévy process has only negative jumps. For the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies for general Lévy processes having both positive and negative jumps. Instead of following the classical approach of using a verification theorem, we validate our solution candidates within a viscosity solution framework. To this end, the value function of our singular control problem is characterized as the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation in the case of general utilities and general Lévy processes.

Journal

Finance and StochasticsSpringer Journals

Published: Oct 1, 2001

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