A Connection between Singular Stochastic Control and Optimal Stopping

Singular Stochastic Control and Optimal Stopping with Partial Information of Itô--Lévy Processes

“…A connection between singular control and optimal stopping for Brownian motion was first established by Karatzas and Shreve [14] and generalized to geometric Brownian motion by Baldursson and Karatzas [5]. This was extended by Boetius and Kohlmann [7], and subsequently extended further by Benth and Reikvam [6], to more general continuous diffusions. More recently, maximum principles for singular stochastic control problems have been studied in [1,2,3,4].…”

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

“…The control process ν t is the cumulative investment made up to time t and c is a general convex cost function. We solve problem (1.1) by relying on the connection existing between singular stochastic control and optimal stopping (see, e.g., [3], [6], [8], [9], [32] and [34]). In fact, we provide the optimal investment strategy ν * in terms of an optimal boundary surface (x, y) → z * (x, y) that splits the state space into action and inaction regions.…”

On an integral equation for the free-boundary of stochastic, irreversible investment problems

“…is the optimal timing problem naturally associated to the optimal investment problem (9). Notice that v(x, y) ≤ 1, for all x ∈ I and y > 0, and that the mapping y → v(x, y) is nonincreasing for any x ∈ I, because π(x, ·) is strictly concave.…”

“…Moreover, the optimal stopping time τ * is such that τ * = inf{t ≥ 0 : ν * (t) > 0}, with ν * the optimal singular control. Later on, this kind of link has been established also for more complicated dynamics of the controlled diffusion; that is the case, for example, of a geometric Brownian motion [2], or of a quite general controlled Itô diffusion (see [9] and [10], among others).…”