Controlled Stochastic Differential Equations under Poisson Uncertainty and with Unbounded Utility
AbstractThe 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. --
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Bibliographic InfoPaper provided by Dresden University of Technology, Faculty of Business and Economics, Department of Economics in its series Dresden Discussion Paper Series in Economics with number 03/05.
Date of creation: 2005
Date of revision:
Stochastic differential equation; Poisson process; Bellman equation;
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- C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
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