PreprintMultiscale LinearQuadratic Stochastic Optimal Control with multiplicative noiseBeniamin Goldys, Gianmario Tessitore, James Yang and Zhou ZhouAbstractWe investigate asymptotic properties of a finitetime horizon stochastic control when the cost functional is quadratic and the state process is given by a two component slowfast system that is linear in drift and and in the Brownian diffusion term. We approach this problem by considering the associated differential Riccati equation and reformulating it as a deterministic singular perturbation problem. Asymptotic properties of this deterministic problem are deduced from a version of the Tikhonov Theorem. We propose two methods to approximate the value function of the stochastic optimal control problem. The first one is by constructing an approximately optimal control process whilst the second one is by identifying the limit to the value function. Both approximation methods rely on a new result on the existence of a solution to a coupled differentialalgebraic Riccati equation with certain stability properties. Keywords: singular perturbations, linearquadratic stochastic optimal control, differentialalgebraic Riccati equation, slowfast system multiscale system.AMS Subject Classification: Primary 34E13; secondary 34D15, 93C70, 93E20.
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