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

Abstract

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 dass 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 dass of numerical methods for the investment-consumption model in question.

Fractal First-Order Partial Differential Equations

“…See also [25,23,1] and more recently [6][7][8]. A general theory for non-linear integro-partial differential equations is developed by Jakobsen and Karlsen [19,20].…”

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

“…In some recent developments [9,10], Barles & Jakobsen used solutions of certain switching systems to generate suitable approximations of the viscosity solution of the Bellman equation associated with the optimal control of diffusion processes. In a future work we will adapt this approach to the nonlocal Bellman equation of controlled jump-diffusion processes, which is drawing a lot of interests these days due to its applications in mathematical finance (see for example [3], [2], [14], [15], [19] and the references therein). To derive error estimates like those in [9,10] for the nonlocal Bellman equation we need to have at our disposal a viscosity solution theory for switching systems of the type (1.1).…”

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

“…An existence result and a comparison principle among uniformly continuous and at most linearly growing viscosity sub-and supersolutions of fully nonlinear parabolic integro-partial differential equations of the Bellman type are proved in [25], see also [21] for some other existence results. The Bellman equations (variational inequalities) associated with some singular stochastic control problems arising in finance are studied in [7,8]. In [16], the authors prove a "non-local" maximum principle for semicontinuous viscosity sub-and supersolutions of integro-partial differential equations, which should be compared with the "local" maximum principle for semicontinuous functions [11].…”

“…Empirical work shows that the normal distribution poorly fits the logreturn data for, e.g., stock prices. Among other things the data show heavier tails than predicted by the normal distribution, and it has in recent years been suggested to model logreturns by generalized hyperbolic distributions (see the references in [6,7,8,10,26,9] for relevant works).…”