arXiv subjects

Zhesheng Liu

Publications and source records attributed to Zhesheng Liu.

2 recordsLinked to original sources

arXiv2025

Singular stochastic control problems motivated by the optimal sustainable exploitation of an ecosystem

We derive the explicit solutions to singular stochastic control problems of the monotone follower type with (a) an expected discounted criterion, (b) an expected ergodic criterion and (c) a pathwise ergodic criterion. These problems have been motivated by the optimal sustainable exploitation of an ecosystem, such as a natural fishery. Under general assumptions on the diffusion coefficients, the discounting rate function, the running payoff function and the marginal profit of control action, we show that the optimal strategies are of a threshold type. We solve the three problems by first constructing suitable solutions to their associated HJB equations, which take the form of quasi-variational inequalities with gradient constraints. In the cases of the ergodic control problems, we also use a suitable new variational argument. Furthermore, we establish the convergence of the solution of the discounted control problem to the one of the ergodic control problems as the discounting rate function tends to 0 in an Abelian sense.

math.OC↗

arXiv2024

The Solution to an Impulse Control Problem Motivated by Optimal Harvesting

We consider a stochastic impulse control problem that is motivated by applications such as the optimal exploitation of a natural resource. In particular, we consider a stochastic system whose uncontrolled state dynamics are modelled by a non-explosive positive linear diffusion. The control that can be applied to this system takes the form of one-sided impulsive action. The objective of the control problem is to maximise a discounted performance criterion that rewards the effect of control action but involves a fixed cost at each time of a control intervention. We derive the complete solution to this problem under general assumptions. It turns out that the solution can take four qualitatively different forms, several of which have not been observed in the literature. In two of the four cases, there exist only $\varepsilon$-optimal control strategies. We also show that the boundary classification of 0 may play a critical role in the solution of the problem. Furthermore, we develop a way for establishing the strong solution to a stochastic impulse control problem's optimally controlled SDE.

math.OC↗