Dynamic programming for a Markov-switching jump–diffusion
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摘要
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.
论文关键词:93E20,49L20,91G10,Stochastic optimal control,Jump–diffusion,Markov-switching,Optimal consumption–investment
论文评审过程:Received 20 August 2013, Revised 21 January 2014, Available online 5 February 2014.
论文官网地址:https://doi.org/10.1016/j.cam.2014.01.021