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Mayer and optimal stopping stochastic control problems with discontinuous cost. (English) Zbl 1215.93150

Summary: We study two classes of stochastic control problems with semicontinuous costs: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.

MSC:

93E20 Optimal stochastic control
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49J40 Variational inequalities
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
49N15 Duality theory (optimization)
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