BFS 2002

Poster Presentation

A Markov chain approximation scheme for an investment-consumption problem with intertemporal substitution and LÚvy driven stock prices

Said Elghanjaoui, Kenneth Hvistendahl Karlsen

We study an optimal investment-consumption problem for a small investor whose wealth is divided between a riskless asset (a bank account) and a risky asset (a stock) with log-returns following a LÚvy process. The investor preferences, in contrast to the standard von Neumann-Morgenstern time-additive preferences, allow for cumulative consumption patterns with possible jumps/singular sample paths and they incorporate the notion of local substitution. The dynamic programming equation of this singular stochastic control problem is a degenerate elliptic integro-differential variational inequality (a free boundary problem). Herein we present and analyze a Markov chain approximation scheme for solving the investment-consumption problem. A feature of the suggested numerical scheme is that it is based on a simplified dynamic programming equation obtained by approximating the original LÚvy process by a more simple and tractable (LÚvy) process which can be written as an independent sum of a drift, a Brownian component, and a finite number of compound Poisson processes. This approximation reduces the integral operator in the dynamic programming equation to a finite series operator. The convergence analysis of the numerical scheme is based on the theory of viscosity solutions.