Stochastic Optimal Control of Systems with Fast and Slow States

We consider the problem of optimal stochastic control of diffusion processes containing "fast" and "slow" dynamics. The systems are considered on an unbounded state space. The analysis highlights the key role played by ergodicity of the fast state variables. We use a stochastic stability theorem of Khas\´minskii to determine the conditions under which ergodicity holds and the optimal control problem is well posed. The limiting control problem obtained as the small parameter goes to zero retains an interesting interdependence between fast and slow variables.