Singular Perturbation of Stochastic Control and Differential Games

The defining trait of singular perturbation problems in dynamical systems is the degeneracy of the highest order differential term when a small parameter is formally set to be zero. The implication is that the limiting solution does not entirely coincide with the solution to the degenerated system. It then becomes apparent that the derivation of the limiting solution is non-trivial. However, under certain circumstances, this has been resolved by the Tikhonov's theorem. On the other hand, these problems arise naturally in slow-fast or multiscale models where the small parameter represents the ratio between the evolutionary speeds. Moreover, the limiting solution is often of lower dimensionality and offers a viable method for dimension reduction. For these reasons, singular perturbation techniques have been widely applied in optimisation theory to disciplines such as ecology, robotics, finance and physics, to name a few. In this thesis, we propose three optimisation problems described by a quadratic cost function and a coupled pair of slow-fast linear state equations driven by Brownian motion. The first one is an optimal control problem when the state and control appear in the diffusion coefficient of the noise. The second one is a two-player zero-sum differential game with a constant diffusion coefficient. And third, is an optimal control problem with processes taking values in infinite dimensional spaces. Our aim is to investigate the limiting behaviour of these optimisation problems and its value functions. The general approach is to convert the stochastic and controlled singular perturbation problem into a classical and deterministic singular perturbation problem via the Riccati equation. Consequently, a version of Tikhonov's theorem has to be formulated