Necessary stochastic maximum principle for dissipative systems on infinite time horizon

Abstract

Abstract. We develop a necessary stochastic maximum principle for a finite-dimensional stochastic control problem in infinite horizon under a polynomial growth and joint monotonicity assumption on the coefficients. The second assumption generalizes the usual one in the sense that it is formulated as a joint condition for the drift and the diffusion term. The main difficulties concern the construction of the first and second order adjoint processes by solving backward equations on an unbounded time interval. The first adjoint process is characterized as a solution to a backward SDE, which is well-posed thanks to a duality argument. The second one can be defined via another duality relation written in terms of the Hamiltonian of the system and linearized state equation. Some known models verifying the joint monotonicity assumption are discussed as well.

Ergodic Maximum Principle for Stochastic Systems

“…(36) which will be estimated by duality. For the approximating equation (36) a wellposedness result has been already adressed in [15]. To shorten the notation in the following paragraphs, let us denote…”

Infinite horizon backward stochastic Volterra integral equations and discounted control problems

“…Infinite horizon stochastic control problems for stochastic differential equations (SDEs for short) with discounted cost functional have been studied in, for example, [11,24,25]. In these papers, infinite horizon BSDEs play central roles in different ways.…”

“…Fuhrman and Tessitore [11] investigated a relationship between an infinite horizon BSDE and an elliptic Hamilton-Jacobi-Bellman equation to obtain the optimal control. Maslowski and Veverka [24] obtained a sufficient maximum principle, and Orrieri and Veverka [25] established a necessary maximum principle for an SDE with dissipative coefficients. Our results are extensions of [24,25] to the Volterra setting.…”

“…Maslowski and Veverka [24] obtained a sufficient maximum principle, and Orrieri and Veverka [25] established a necessary maximum principle for an SDE with dissipative coefficients. Our results are extensions of [24,25] to the Volterra setting. Indeed, in the SDEs setting, the adjoint equation which we obtain in the general Volterra setting is reduced to the infinite horizon BSDE considered in the aforementioned papers (see Example 4.9).…”

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

“…Based on the motivation of the above literature, the authors find that the few of the works focused on optimal control of infinite‐horizon mean‐field stochastic differential games (see References 20‐23 and references therein). Infinite‐horizon optimal control problems arise naturally in economics when dealing with dynamical models of optimal allocation of resources.…”