A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

Abstract

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward-backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzerosum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.Keywords Forward-backward stochastic differential equations · Markov regimeswitching · Stochastic differential games · Optimal investment · Stochastic maximum principleThe project on which this publication is based has been carried out with funding

Risk‐sensitive maximum principle for stochastic optimal control of mean‐field type Markov regime‐switching jump‐diffusion systems

“…The stochastic maximum principle type weak necessary condition of optimality for a general class of SDEs of Markov regime‐switching jump‐diffusion models was established by Li and Zheng 11 using Clarke's generalized gradient of the Hamiltonian. The work by Menoukeu‐Pamen and Momeya 12 considered the stochastic maximum principle for forward–backward SDEs. The stochastic control problem for SDEs of Markov regime‐switching jump‐diffusion models with delay was studied by Menoukeu‐Pamen 13 and Savku and Weber 14 …”

A stochastic control approach for constrained stochastic differential games with jumps and regimes

“…Additionally, we can combine regime-switches with stochastic optimal control, which is another fundamental method of managing random events (for complete treatments of control theory, see [2] and [3]). Hence, these models have attracted many researchers so far such as [4][5][6][7][8][9][10][11][12].…”

Optimal control of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward system with Lévy processes

“…From the empirical point of view, Markov regime‐switching models perform better than the traditional models based on the diffusion processes. In particular, Pamen and Momeya 18 studied the optimal control problem for stochastic differential games under Markov regime‐switching FBSDE with jumps. Moreover, stochastic control problems related to Lvy processes associated with Teugels martingales are studied by various authors (see References 10,19).…”

“…Infinite‐horizon optimal control problems arise naturally in economics when dealing with dynamical models of optimal allocation of resources. Thus, the main contributions in this article is listed as, Comparing with existing work of Markov regime‐switching FBSDE in Reference 18, the proposed work is generalized to infinite‐horizon case of time and perturbed with Lvy processes. The infinite‐horizon version of stochastic maximum principle and necessary conditions for optimality are established using the transversality conditions and assumption of convex control domain. In classical stochastic optimal control problem, there is a single control u ( t ) which corresponds to the single objective functional to be optimized. Rather, the stochastic optimal control of nonzero sum differential game have two controls, namely, u 1 ( t ), u 2 ( t ) and corresponding to two objective functional for each player, whereas each player attempts to control the state of the system so as to achieve the desired goal.…”