Abstract
This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. First, a general sufficient maximum principle for optimal control for a system, driven by a Markov regime-switching forward–backward jump–diffusion model, is developed. In the regime-switching case, it might happen that the associated Hamiltonian is not concave and hence the classical maximum principle cannot be applied. Hence, an equivalent type maximum principle is introduced and proved. In view of solving an optimal control problem when the Hamiltonian is not concave, we use a third approach based on Malliavin calculus to derive a general stochastic maximum principle. This approach also enables us to derive an explicit solution of a control problem when the concavity assumption is not satisfied. In addition, the framework we propose allows us to apply our results to solve a recursive utility maximization problem.
Highlights
Optimal control problem for Markovian regime-switching model has received a lot of attention recently; See, e.g., [7, 8, 16, 24, 26]
One of the reasons for looking at regime switching in finance for example is that, they enable to capture exogenous macroeconomic cycles against which asset prices evolve There are two existing approaches to solve stochastic optimal control problem in the literature: The dynamic programing and the stochastic maximum principle
The stochastic maximum principle is given in terms of an adjoint equation, which is solution to a backward stochastic differential equation (BSDE)
Summary
Optimal control problem for Markovian regime-switching model has received a lot of attention recently; See, e.g., [7, 8, 16, 24, 26]. Forward-backward stochastic differential equations, Malliavin calculus, regime switching, recursive utility maximization, stochastic maximum principle. We are able to solve an optimal control problem with non concave utility function for Markov regime-switching jumps-diffusion based on Malliavin calculus. This paper discusses a partial information stochastic maximum principle for optimal control of forward backward stochastic differential equation (FBSDE) driven by Markov regimeswitching jump-diffusion process. A critical point for the performance functional of a partial information FBSDE problem is a conditional critical point for the associated Hamiltonian and vice versa The proof of such equivalent maximum principle requires the use of some variational equations (compare with [23, Section 4]). A problem of recursive utility maximization with Markovian regime-switching is studied
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