  We investigate the asymptotic properties of a finite-time horizon
linear-quadratic optimal control problem driven by a multiscale stochastic
process with multiplicative Brownian noise. We approach the problem by
considering the associated differential Riccati equation and reformulating it
as a classical and deterministic singular perturbation problem. Asymptotic
properties of this deterministic problem can be gathered from the well-known
Tikhonov Theorem. Consequently, we are able to propose two approximation
methods to the value function of the stochastic optimal control problem. The
first is by constructing an approximately optimal control process whilst the
second is by finding the direct limit to the value function. Both approximation
methods rely on the existence of a solution to a coupled differential-algebraic
Riccati equation with certain stability properties - this is the main
difficulty of the paper.
