Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk (2306.01998v2)

Abstract: Environmental management and restoration should be designed such that the risk and uncertainty owing to nonlinear stochastic systems can be successfully addressed. We apply the robustified dynamic Orlicz risk to the modeling and analysis of environmental management and restoration to consider both the risk and uncertainty within a unified theory. We focus on the control of a jump-driven hybrid stochastic system that represents macrophyte dynamics. The dynamic programming equation based on the Orlicz risk is first obtained heuristically, from which the associated Hamilton-Jacobi-BeLLMan (HJB) equation is derived. In the proposed Orlicz risk, the risk aversion of the decision-maker is represented by a power coefficient that resembles a certainty equivalence, whereas the uncertainty aversion is represented by the Kullback-Leibler divergence, in which the risk and uncertainty are handled consistently and separately. The HJB equation includes a new state-dependent discount factor that arises from the uncertainty aversion, which leads to a unique, nonlinear, and nonlocal term. The link between the proposed and classical stochastic control problems is discussed with a focus on control-dependent discount rates. We propose a finite difference method for computing the HJB equation. Finally, the proposed model is applied to an optimal harvesting problem for macrophytes in a brackish lake that contains both growing and drifting populations.

• The paper presents a robustified dynamic Orlicz risk measure that integrates risk and uncertainty aversion for improved environmental management.
• The methodology employs hybrid stochastic differential equations and a nonlinear Hamilton–Jacobi–Bellman equation to model macrophyte dynamics.
• Results demonstrate the model’s effectiveness in tailoring intervention policies across diverse ecological conditions for optimal restoration.

Overview of "Environmental Management and Restoration under Unified Risk and Uncertainty Using Robustified Dynamic Orlicz Risk"

The paper investigates a novel approach to environmental management and restoration in contexts characterized by nonlinear stochastic systems and incomplete data. The authors, Hidekazu Yoshioka, Motoh Tsujimura, Futoshi Aranishi, and Tomomi Tanaka, present a robust analytical method leveraging the dynamic Orlicz risk, expanding its utility to model risk and uncertainty within environmental systems effectively. This innovative application is particularly focused on managing the macrophyte dynamics in aquatic environments through optimal control methods.

Methodology

The paper introduces a mathematical framework built around hybrid stochastic differential equations (SDEs) influenced by jump processes and intended to model the dynamics of macrophyte populations. The authors develop a robustified dynamic Orlicz risk measure to encapsulate both risk aversion, through a certainty equivalence expressed by a power coefficient, and uncertainty aversion, captured by the Kullback–Leibler divergence. The framework also involves a distinctive nonlinear Hamilton–Jacobi–BeLLMan (HJB) equation incorporating a state-dependent discount rate arising from uncertainty aversion.

Their approach uses this framework to handle key challenges in environmental management, such as macrophyte growth in a brackish lake, which includes both stable and less predictable elements. A finite difference method is proposed to solve the HJB equation, enabling numerical assessment and implementation of an optimal policy.

Implications and Results

The application of this methodological advancement to the environmental context signifies a shift towards integrating robust statistical measures in ecological management. Practically, their approach offers a toolset for decision-makers to evaluate intervention strategies amidst uncertain environmental dynamics. This model is particularly beneficial in cases like the management and restoration of aquatic vegetation, where stochastic events and incomplete knowledge of the system dynamics complicate decision-making.

In their empirical application, significant variability was observed between different survey stations with respect to optimal intervention policies, indicating the model's sensitivity and adaptability to varying conditions. This showcases the utility of the robust dynamic Orlicz risk model in tailoring strategies that respect local ecological dynamics and socio-economic considerations alternatively.

Future Directions

The paper suggests several future research pathways, including the adaptation of the proposed Orlicz risk measure to other environmental and sustainability challenges, like tourism and climate-driven socio-economic changes. Additionally, exploring other divergence measures beyond the Kullback–Leibler divergence presents an opportunity for refining model sensitivity and operational applicability. The authors also hint at the potential of leveraging this framework to address problems with unbounded state dynamics, suggesting a broadened scope of application in strategic ecological management and predictive analytics.

Conclusion

This paper contributes to fiscal policy and environmental sustainability by integrating robust risk management methods into stochastic control problems. The distinct formulation of the HJB equation with a state-dependent discount rate offers a new perspective for managing ecological risks under uncertainty, enriching both theoretical insights and practical applications in environmental science. The paper underscores the necessity for interdisciplinary approaches and advanced quantitative methods in resolving complex decision-making problems associated with environmental management and restoration.