Stochastic time-optimal control problems

Two types of stochastic time-optimal controls in a one-dimensional setting are considered. Multidimensional problems, in the case of complete state information available and the system modelled by stochastic differential equations, are studied under the formulation of minimising the expected transient-response time. The necessary condition of optimality is the satisfaction for the value function of a parabolic partial differential equation with boundary conditions. The sufficient condition of optimality is also provided, based on Dynkin´s formula. Finally, three examples are given