Necessary stochastic maximum principle for dissipative systems on infinite time horizon

TL;DR

This paper establishes a necessary stochastic maximum principle for finite-dimensional dissipative systems on an infinite time horizon, addressing challenges in constructing adjoint processes under polynomial growth and joint monotonicity conditions.

Contribution

It introduces a generalized joint monotonicity condition for coefficients and develops a method to construct adjoint processes over an unbounded time interval.

Findings

Successfully characterizes the first adjoint process via backward SDE.

Defines the second adjoint process through duality relations involving the Hamiltonian.

Discusses models satisfying the joint monotonicity assumption.

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.