Dynamic programming for a Markov-switching jump�diffusion

• Autores: N. Azevedo, Diogo Pinheiro, G.- W. Weber
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 267, Nº 1, 2014, págs. 1-19
• Idioma: inglés
• DOI: 10.1016/j.cam.2014.01.021
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• Resumen

  • 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 Lévy 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.