Hamilton-Jacobi-Bellman-Isaacs equation for rational inattention in the long-run management of river environments under uncertainty

TL;DR

This paper develops a stochastic control model for long-term river environmental management, incorporating rational inattention and uncertainty, and provides numerical methods and applications demonstrating the benefits of adaptive, state-dependent strategies.

Contribution

It introduces a novel stochastic control framework with a Hamilton-Jacobi-Bellman-Isaacs equation for river management under uncertainty, incorporating rational inattention via Erlangization.

Findings

Rational inattention improves environmental management outcomes.

The proposed numerical scheme guarantees solution stability and accuracy.

Model applications show adaptive strategies outperform static approaches.

Abstract

A new stochastic control model for the long-run environmental management of rivers is mathematically and numerically analyzed, focusing on a modern sediment replenishment problem with unique nonsmooth and nonlinear properties. Rational inattention as a novel adaptive strategy to collect information and intervene against the target system is modeled using Erlangization. The system dynamics containing the river discharge following a continuous-state branching with an immigration-type process and the controlled sediment storage dynamics lead to a nonsmooth and nonlocal infinitesimal generator. Modeling uncertainty, which is ubiquitous in certain applications, is considered in a robust control framework in which deviations between the benchmark and distorted models are penalized through relative entropy. The partial integro-differential Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation as an optimality equation is derived, and its uniqueness, existence, and optimality are discussed. A monotone finite difference scheme guaranteeing the boundedness and uniqueness of numerical solutions is proposed to discretize the HJBI equation and is verified based on manufactured solutions. Model applications are also conducted with the parameter values identified from the available data and physical formulae. The computational results suggest that environmental management should be rationally inattentive in a state-dependent and adaptive manner.