Bounded Variation Singular Stochastic Control and Associated Dynkin Game

Abstract

Motivated by theobservation—yet present in derivations of the Black-Scholes formula—that the value of a dynamic optimization problem is actually a stochastic process determined by the future development of the stochastic system, we reformulate a singular control problem as optimization problem for decoupled forward-backward SDE. The controlled process takes the form of a general Brownian diffusion in one dimension, with bounded variation control.

Applications are given in the field of real investment and (real) options; many other optimization problems such as hedging with transaction costs or dividend control also exhibit a singular behaviour. Our reformulation enables us to consider nonadditive or stochastic differential utilities in the control problem, and exhibits formal similarities with reflected backward SDE and g-semisolutions.

Related to the control problem we find a two-player stochastic game of optimal stopping (or Dynkin game). We establish a well-known relation between both problems for this general situation, namely that the partial derivative of the value of the control problem $\frac{\partial } {{\partial x}}V$ equals the value of the Dynkin game u, and a saddle point for this game can be derived from an optimal control in the control problem. The arguments are based on stochastic analysis, especially comparison theorems for the solutions of controlled forward and backward stochastic differential equations.