Controlled Stochastic Differential Equations under Poisson Uncertainty and with Unbounded Utility

Ken Sennewald

No 03/05, Dresden Discussion Paper Series in Economics from Technische Universität Dresden, Faculty of Business and Economics, Department of Economics

Abstract: The present paper is concerned with the optimal control of stochastic differential equations, where uncertainty stems from one or more independent Poisson processes. Optimal behavior in such a setup (e.g., optimal consumption) is usually determined by employing the Hamilton-Jacobi-Bellman equation. This, however, requires strong assumptions on the model, such as a bounded utility function and bounded coefficients in the controlled differential equation. The present paper relaxes these assumptions. We show that one can still use the Hamilton-Jacobi-Bellman equation as a necessary criterion for optimality if the utility function and the coefficients are linearly bounded. We also derive sufficiency in a verification theorem without imposing any boundedness condition at all. It is finally shown that, under very mild assumptions, an optimal Markov control is optimal even within the class of general controls.

Keywords: Stochastic differential equation; Poisson process; Bellman equation (search for similar items in EconPapers)
JEL-codes: C61 (search for similar items in EconPapers)
Date: 2005
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Citations: View citations in EconPapers (4)

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