Singular Stochastic Control and Optimal Stopping

SIAM Journal on Control and Optimization, 2004

In this paper, we study an optimal singular stochastic control problem. By using a time transformation, this problem is shown to be equivalent to an auxiliary control problem defined as a combination of an optimal stopping problem and a classical control problem. For this auxiliary control problem, the controller must choose a stopping time (optimal stopping), and the new control variables belong to a compact set. This equivalence is obtained by showing that the (discontinuous) state process governed by a singular control is given by a time transformation of an auxiliary state process governed by a classical bounded control. It is proved that the value functions for these two problems are equal. For a general form of the cost, the existence of an optimal singular control is established under certain technical hypotheses. Moreover, the problem of approximating singular optimal control by absolutely continuous controls is discussed in the same class of admissible controls.

Finance and Stochastics, 1996

SSRN Electronic Journal, 2000

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem.

Mathematics of Operations Research, 2008

Mathematics of Operations Research, 2008

Motivated by the analysis of financial instruments with multiple exercise rights of American type and mean reverting underlyers, we formulate and solve the optimal multiple-stopping problem for a general linear regular diffusion process and a general reward function. Instead of relying on specific properties of geometric Brownian motion and call and put option payoffs as in most of the existing literature, we use general theory of optimal stopping for diffusions, and we illustrate the resulting optimal exercise policies by concrete examples and constructive recipes.

The Annals of Applied Probability, 2008

A singular stochastic control problem with state constraints in two-dimensions is studied. We show that the value function is C 1 and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem, we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133-2171] on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not C 1 . This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is C 2 in the interior of the no-action region. We then use a verification argument applied to a suitable C 2 approximation of the value function to establish optimality of the conjectured control.

The Annals of Applied Probability, 2006

We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems \citehar, associated with a broad family of stochastic networks, follows as a consequence.

This paper examines a class of singular stochastic control problems with convex objective functions. In Section 2, we use tools from convex analysis to derive necessary and sufficient first order conditions for this class of optimisation problems. The main result of this paper is Theorem 9 which uses results from optimal stopping to establish the link between singular stochastic control and Gittin's index without the need to appeal to the representation result in [5]. In Sections 3-5 we assume the singular control problem is driven by a L\'{e}vy process. Expressions for the Gittin's index are derived in terms of the Wiener-Hopf factorisation. This allows us to broaden the class of parameterised optimal stopping problems with explicit solutions examined in [6] and derive explicit solutions to the singular control problems studied in Section 2. In Section 4, we apply our results to the `monotone follower' problem which originates in [7] and [28]. In Section 5, we apply our results to an irreversible investment problem which has been studied in [9], [35] and [40].

SSRN Electronic Journal, 2000

In this paper we provide a complete theoretical analysis of a two-dimensional degenerate non convex singular stochastic control problem. The optimisation is motivated by a storage-consumption model in an electricity market, and features a stochastic real-valued spot price modelled by Brownian motion. We find analytical expressions for the value function, the optimal control and the boundaries of the action and inaction regions. The optimal policy is characterised in terms of two monotone and discontinuous repelling free boundaries, although part of one boundary is constant and and the smooth fit condition holds there.