A class of solvable singular stochastic control problems

Abstract

We consider the determination of the optimal stationary singular stochastic control of a linear diffusion for a class of average cumulative cost minimization problems arising in various financial and economic applications of stochastic control theory. We present a set of conditions under which the optimal policy is of the standard local time reflection type and state the first order conditions from which the boundaries can be determined. Since the conditions do not require symmetry or convexity of the costs, our results cover also the cases where costs are asymmetric and non-convex. We also investigate the comparative static properties of the optimal policy and delineate circumstances under which higher volatility expands the continuation region where utilizing the control is suboptimal.

Optimal entry to an irreversible investment plan with non convex costs

“…Proof of (ii). To prove (ii) we simply argue as in Theorem 5.5, concluding that V (x, c) = V (1) * (x, c), x ∈ R and (1) * solves (5.18) with B = (−∞, x 0 1 (c)).…”

“…Let us assume now that 2 < (2) * , then taking x ∈ ( 2 , (2) * ), applying the Itô-Tanaka formula until the first exit time from the open set ( (1) * , (2) * ) and using arguments similar to those in Steps 1. and 2. we end up with a contradiction. Hence 2 = (2) * ; analogous arguments can be applied to establish that 1 = (1) * .…”

“…From Remark 5.4 this implies that part iii)(a) of Proposition 5.2 holds. The optimal entry policy is then of threshold type, and the continuation region has the lower boundary (1) * (c). Further, once the contract is entered, the optimal purchasing policy is then to wait until the price X is above the level γ * (c), at which time the inventory is instantaneously filled.…”

“…We fix c ∈ [0, 1), assume that there exists a triple { 1 , 2 , 3 } = { (1) * , (2) * , (3) * } solving (5.23)-(5.25) and define a stopping time…”

“…This control is assumed to be monotone, so that the sale of electricity back to the market is not possible, and also bounded to reflect the fact that the inventory used for storage has finite capacity. The investment problem falls into the class of singular stochastic control (SSC) problems (see [1,15,16], among others).…”