Special Semester on Stochastics with Emphasis on Finance
Linz, September 2008 - December 2008
Presentation: A problem of optimal portfolio/consumption choice in a liquidity risk model with random trading times

Workshop: Kick-off-Workshop

Time: Thu, September 11, 2008, 10:50-11:40

Speaker: Huyên Pham


We consider a portfolio/consumption choice problem in a market model with liquidity risk. The main feature is that the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption.
This is a mixed discrete/continuous stochastic control problem, nonstandard in the literature. We first show how the dynamic programming principle leads to a coupled system of Integro-Differential Equations (IDE), and we prove an analytic characterization of this control problem by adapting the concept of viscosity solutions. Next, we derive smoothness results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a second-order ODE.
In the numerical part of this work, we provide a convergent numerical algorithm for the resolution to this coupled system of IDE. Several numerical experiments illustrate the impact of the restricted liquidity trading opportunities, and we measure in particular the utility loss with respect to the classical Merton consumption problem. We finally illustrate the behavior of optimal consumption policies between two trading dates.

Based on joint works with: P. Tankov, F. Gozzi and A. Cretarola.

Presentation slides (pdf, 190 KB)

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