  Control of continuous time dynamics with multiplicative noise is a classic
topic in stochastic optimal control. This work addresses the problem of
designing infinite horizon optimal controls with stability guarantees for
\textit{a single agent or large population systems} of identical,
non-cooperative and non-networked agents, with multi-dimensional and nonlinear
stochastic dynamics excited by multiplicative noise. For agent dynamics
belonging to the class of reversible diffusion processes, we provide
constraints on the state and control cost functions which guarantee stability
of the closed-loop system under the action of the individual optimal controls.
A condition relating the state-dependent control cost and volatility is
introduced to prove the stability of the equilibrium density. This condition is
a special case of the constraint required to use the path integral Feynman-Kac
formula for computing the control. We investigate the connection between the
stabilizing optimal control and the path integral formalism, leading us to a
control law formulation expressed exclusively in terms of the desired
equilibrium density.
