Regularity properties in a state-constrained expected utility maximization problem

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Abstract

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of an optimal strategy under rather mild model assumptions. This will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove a dynamic programming principle without appealing to the classical measurable selection arguments. This permits us to establish a tight relation between our value function and a nonlinear parabolic degenerated Hamilton–Jacobi–Bellman (HJB) equation with singularity. To conclude, we show a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.

Keywords

Expected utility maximization problem Value function Price impact Optimal strategy Dynamic programming principle Bellman’s principle Hamilton–Jacobi–Bellman equation Viscosity solution Comparison principle 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MannheimMannheimGermany

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