A model of optimal consumption under liquidity risk with random trading times and its coupled system of integrodifferential equations

Huyên Pham (1, 2) , Peter Tankov (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 CREST - Centre de Recherche en Économie et Statistique

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 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. We also provide a convergent numerical algorithm for the resolution to this coupled system of IDE, and illustrate our results with some numerical experiments.

Keywords: liquidity; portfolio/consumption problem; integrodifferential equations; viscosity solutions

Domains: Probability [math.PR]

Huyên Pham, Peter Tankov. A model of optimal consumption under liquidity risk with random trading times and its coupled system of integrodifferential equations. 2006. ⟨hal-00090149⟩