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In memory of Monroe D. Donsker who showed me the wonderful world of Wiener integrals. AbstractThe large deviation problem of Wentzell-Freidlin type is considered for a class of semilinear parabolic equations perturbed by a small multiplicative noise. In a Hilbert space setting, it is proved that, under suitable conditions, the associated family of solution measures, depending on a small parameter, obeys the large deviation principle with respect to a certain rate function.

Abstract. We consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients. The control acts additively on the state of the system. Our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process. It is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space-time domain. By using some decomposition arguments, the problems of a two-sided control, i.e. optimal corrections, and the case with constraints on the resources, i.e. finite fuel, can be reduced to a simpler case of only one-sided control, i.e. a monotone follower. These results are applied to solving some examples by the so-called method of similarity solutions.

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