A Coupled System of Integrodifferential Equations Arising in Liquidity Risk Model |
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Authors: | Huyên Pham Peter Tankov |
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Institution: | (1) Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, Universités Paris 6–Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France;(2) CREST and Institut Universitaire de France, Paris, France |
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Abstract: | We study the mathematical aspects of the portfolio/consumption choice problem in a market model with liquidity risk introduced
in (Pham and Tankov, Math. Finance, 2006, to appear). In this model, 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 time 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. This coupled system of IDE may be numerically solved
by a decoupling algorithm, and this is the topic of a companion paper (Pham and Tankov, Math. Finance, 2006, to appear). |
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Keywords: | Liquidity Portfolio/consumption problem Integrodifferential equations Viscosity solutions Comparison principle |
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