On a class of singular stochastic control problems for reflected diffusions

Abstract

Reflected diffusions naturally arise in many problems from applications ranging from economics and mathematical biology to queueing theory. In this paper we consider a class of infinite time-horizon singular stochastic control problems for a general onedimensional diffusion that is reflected at zero. We assume that exerting control leads to a state-dependent instantaneous reward, whereas reflecting the diffusion at zero gives rise to a proportional cost with constant marginal value. The aim is to maximize the total expected reward, minus the total expected cost of reflection. We show that depending on the properties of the state-dependent instantaneous reward we can have qualitatively different kinds of optimal strategies. The techniques employed are those of stochastic control and of the theory of linear diffusions.

An Optimal Dividend Problem with Capital Injections over a Finite Horizon

“…Problem (5.1) is perhaps the most common formulation of the optimal dividend problem with capital injections (see, e.g., Kulenko and Schmidli [24], Lokka and Zervos [26], Zhu and Yang [35] and references therein). However, to the best of our knowledge, no previous work has considered such a problem in the case of a finite time horizon, whereas problem (5.1) has been extensively studied when T = +∞ (see, e.g., Ferrari [16] and references therein). In particular, it has been shown, e.g., in [16] that in the case T = +∞ the optimal dividend strategy is triggered by a boundary b ∞ > 0 that can be characterized as the solution to a nonlinear algebraic equation (see Proposition 3.2 in [16]).…”

“…However, to the best of our knowledge, no previous work has considered such a problem in the case of a finite time horizon, whereas problem (5.1) has been extensively studied when T = +∞ (see, e.g., Ferrari [16] and references therein). In particular, it has been shown, e.g., in [16] that in the case T = +∞ the optimal dividend strategy is triggered by a boundary b ∞ > 0 that can be characterized as the solution to a nonlinear algebraic equation (see Proposition 3.2 in [16]). In Proposition 3.6 of [16] such a trigger value is also shown to be the optimal stopping boundary of problem (5.4) below (when the optimization is performed over all the F-stopping times).…”

“…In particular, it has been shown, e.g., in [16] that in the case T = +∞ the optimal dividend strategy is triggered by a boundary b ∞ > 0 that can be characterized as the solution to a nonlinear algebraic equation (see Proposition 3.2 in [16]). In Proposition 3.6 of [16] such a trigger value is also shown to be the optimal stopping boundary of problem (5.4) below (when the optimization is performed over all the F-stopping times).…”

“…In Lokka and Zervos [26] the shareholders can choose the capital injections' policy and, in absence of any interventions, the surplus process follows a Brownian motion with drift. Other works in which the surplus process evolves as a general one-dimensional diffusion are the ones by Ferrari [16], Zhu and Yang [35], and Shreve et al [34]. Optimal dividends and capital injections in a jump-diffusion setting are determined by Avanzi et al in [3].…”

Stochastic recursive optimal control problem with obstacle constraint involving diffusion type control

“…Different from the optimal control problem studied by Ma and Yong [19], this paper allows for a time variant coefficient of the diffusion type control in state equation and generalizes the cost functional to the solution of a RBSDE involving diffusion type control. Ferrari [25] introduced a kind of stochastic optimal control problem with reflected forward state equation involving singular control and expectation utility. In contrast, we consider non-reflected stochastic state equation involving diffusion type control and recursive utility with obstacle constraint.…”