  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.
