HJB and Fokker-Planck equations for river environmental management based on stochastic impulse control with discrete and random observation

Analytical and numerical solutions to ergodic control problems arising in environmental management

“…We work on a usual complete probability space (e.g., Wang 41 ). Key ingredients of the sediment replenishment problem are a sediment storage process as a non‐negative and non‐increasing continuous‐time process and an intervention policy to replenish sediment 35 . See, also Figure 1.…”

“…This is a new optimization problem that has not been addressed so far despite their importance. We consider that sediment storage follows a non‐smooth deteriorating process whose deterioration, decrease of the sediment storage, is driven by deterministic and jump processes 33–35 . The former represents the gradual decrease by base river flows, while the latter represents sudden decrease by floods 36 .…”

“…We assume that the driving jump process is a Lévy process having infinite activities 39 to describe generic river flows in a simple way. This is a critical difference between the earlier models where only the compound Poisson noises were considered 34,35 . Sediment transport is a complex physical process whose complete physical description is still unclear and is often intermittent and fractional, 40 motivating us to use the versatile jump processes having infinite activities.…”

“…We consider that sediment storage follows a non-smooth deteriorating process whose deterioration, decrease of the sediment storage, is driven by deterministic and jump processes. [33][34][35] The former represents the gradual decrease by base river flows, while the latter represents sudden decrease by floods. 36 The non-smooth property comes from a constraint that the sediment storage should be a non-negative physical variable.…”

“…This is a critical difference between the earlier models where only the compound Poisson noises were considered. 34,35 Sediment transport is a complex physical process whose complete physical description is still unclear and is often intermittent and fractional, 40 motivating us to use the versatile jump processes having infinite activities. Sediment replenishment problems with Lévy processes having infinite activities as flexible models have not been considered so far.…”