We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases.

中文翻译：

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马尔可夫体制切换跳跃扩散过程的最佳投资组合和消耗

我们考虑具有状态切换的跳跃扩散过程的最优投资组合和消费问题。在最大化消费的期望折现总效用的准则下，采用动态规划原理和随机极大值原理两种方法求出一般目标函数的最优结果，即对一个偏系统的解。微分方程。此外，我们以电力公用事业为例进行了调查，并分析了最优解的存在性和唯一性。在不卖空和非负消费的约束下，导出了最优策略和价值函数的封闭表达式。此外，还对跳跃扩散模型和纯扩散模型的最优结果进行了比较。