Dynamic programming for a Markov-switching jump-diffusion

Azevedo, N.
Pinheiro, D.
Weber, Gerhard Wilhelm
We consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a Markov-switching jump-diffusion stochastic differential equation. Our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems. More specifically, we provide a detailed proof of Bellman's optimality principle (or dynamic programming principle) and obtain the corresponding Hamilton-Jacobi-Belman equation, which turns out to be a partial integro-differential equation due to the extra terms arising from the Levy process and the Markov process. As an application of our results, we study a finite horizon consumption-investment problem for a jump-diffusion financial market consisting of one risk-free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state Markov process. We provide a detailed study of the optimal strategies for this problem, for the economically relevant families of power utilities and logarithmic utilities.


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Citation Formats
N. Azevedo, D. Pinheiro, and G. W. Weber, “Dynamic programming for a Markov-switching jump-diffusion,” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, pp. 1–19, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/52373.