Singular stochastic control and its relations to Dynkin game and entry-exit problems

Abstract

We consider a bounded variation singular stochastic control problemwith value V, the associated Dynkin game with value u and anassociated entry-exit or optimal switching problem. We establish the relationdV/dx=u known from control of Bronwian motion for a generalsituation with control of a diffusion and a nonlinear cost functionaldefined as solution to a BSDE. A saddle point for the Dynkin game isgiven by the pair of first action times of an optimal control.Through an impulse control approximation scheme we construct asolution to the control problem from solutions to the entry-exitproblem, and obtain an integral representation for the value V. Asa special case we deduce equivalence of monotone control and optimalstopping.In a Markovian setting we characterize the value of the controlproblem in n dimensions as the largest viscosity solution to aquasilinear Hamilton-Jacobi-Bellman PDE with gradient constraints. Dueto the gradient constraints, the latter has no unique solution ingeneral.The methods are from stochastic analysis and include a priori estimates, pathwiseconstruction,comparison theorems for FSDE and BSDE, Ito formula for convexfunctions and nonlinear Feynman-Kacformulae. Using this approach we can drop the condition of a``proper'' operator in the HJB PDEand alter the standard path for comparison towards a global argument.