MPS-RR 2000-15

April 2000

# Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

by:

### Fred Espen Benth, Kenneth Hvistendahl Karlsen, Kristin Reikvam

We investigate an infinite horizon investment-consumption model in which a single
agent consumes and distributes her wealth between a risk-free asset (bank account) and several
risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose
that transactions between the assets incur a transaction cost proportional to the size of the
transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps
intertemporal preferences. This portfolio optimization problem is formulated as a singular
stochastic control problem and is solved using dynamic programming and the theory of viscosity
solutions. The associated dynamic programming equation is a second order degenerate elliptic
integro-differential variational inequality subject to a state constraint boundary condition. The
main result is a characterization of the value function as the unique constrained viscosity solution
of the dynamic programming equation. Emphasis is put on providing a framework that allows
for a general class of Lévy processes. Owing to the complexity of our investment-consumption
model, it is not possible to derive closed form solutions for the value function. Hence the optimal
policies cannot be obtained in closed form from the first order conditions for the dynamic
programming equation. Therefore we have to resort to numerical methods for computing the
value function as well as the associated optimal policies. In view of the viscosity solution theory,
the analysis found in this paper will ensure the convergence of a large class of numerical methods
for the investment-consumption model in question.

Availability: [ gzipped `ps`

-file ] [ `pdf`

-file ]

[ Help on down-loading/viewing/printing ]

This paper has now been published in *Stoch. Stoch. Rep. 74, 517--569 (2002)*