Time‐average stochastic control based on a singular local Lévy model for environmental project planning under habit formation

Abstract

This study applies the theory of stochastic control to an environmental project planning to counteract against the sediment starvation problem in river environments. This can be considered as a time‐average inventory problem to time‐discretely control a continuous‐time system driven by a non‐smooth jump process under habit formation disturbing project implementation. The system is modeled such that the sediment storage dynamics are physically consistent with certain experimental results. Further, the habit formation is modeled as simple linear dynamics and serves as a constraint related to the replenishment amount of the sediment. We show that the time‐average control problem is not necessarily ergodic. Consequently, the effective Hamiltonian may become a non‐constant. Thereafter, ratcheting cases as extreme cases of the irreversible habit formation are considered, owing to them being unique exactly solvable non‐ergodic control problems. The optimality equation associated with a regularized and hence well‐defined control problem is verified. Furthermore, a finite difference scheme is examined against the exactly solvable case and then applied to more complicated cases.