Necessary stochastic maximum principle for dissipative systems on infinite time horizon^∗,∗∗

¹ Dipartimento di Matematica, Università di Pavia. via Ferrata 1, 27100 Pavia, Italia
carlo.orrieri01@ateneopv.it
² Dipartimento di Matematica, Politecnico di Milano. via Bonardi 9, 20133 Milano, Italia
petr.veverka@polimi.it

Received: 26 March 2015
Revised: 20 October 2015
Accepted: 5 December 2015

Abstract

We develop a 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.