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Joint Stochastic Optimal Control and Stopping in Aquaculture: Finite-Difference and PINN-Based Approaches

Kevin Kamm

TL;DR

This work addresses a joint stochastic control and stopping problem in aquaculture, formulating the objective as $J(0,x;u,\tau)$ under stochastic price dynamics. It develops two numerical routes: a finite-difference solver serving as a benchmark for a five-dimensional HJB variational inequality with a free boundary, and a Physics-Informed Neural Network (PINN) method augmented by DeepOS to learn stopping regions. The results show that joint optimization of feeding and harvesting outperforms strategies that optimize only control or stopping, with the PINN approach achieving comparable accuracy to the finite-difference solver while offering scalability to higher dimensions. The combination of PINNs and DeepOS mitigates deficiencies in learning the free boundary, making the approach viable for complex, high-dimensional JCtrlOS problems in finance and economics beyond the aquaculture setting.

Abstract

This paper studies a joint stochastic optimal control and stopping (JCtrlOS) problem motivated by aquaculture operations, where the objective is to maximize farm profit through an optimal feeding strategy and harvesting time under stochastic price dynamics. We introduce a simplified aquaculture model capturing essential biological and economic features, distinguishing between biologically optimal and economically optimal feeding strategies. The problem is formulated as a Hamilton-Jacobi-Bellman variational inequality and corresponding free boundary problem. We develop two numerical solution approaches: First, a finite difference scheme that serves as a benchmark, and second, a Physics-Informed Neural Network (PINN)-based method, combined with a deep optimal stopping (DeepOS) algorithm to improve stopping time accuracy. Numerical experiments demonstrate that while finite differences perform well in medium-dimensional settings, the PINN approach achieves comparable accuracy and is more scalable to higher dimensions where grid-based methods become infeasible. The results confirm that jointly optimizing feeding and harvesting decisions outperforms strategies that neglect either control or stopping.

Joint Stochastic Optimal Control and Stopping in Aquaculture: Finite-Difference and PINN-Based Approaches

TL;DR

This work addresses a joint stochastic control and stopping problem in aquaculture, formulating the objective as under stochastic price dynamics. It develops two numerical routes: a finite-difference solver serving as a benchmark for a five-dimensional HJB variational inequality with a free boundary, and a Physics-Informed Neural Network (PINN) method augmented by DeepOS to learn stopping regions. The results show that joint optimization of feeding and harvesting outperforms strategies that optimize only control or stopping, with the PINN approach achieving comparable accuracy to the finite-difference solver while offering scalability to higher dimensions. The combination of PINNs and DeepOS mitigates deficiencies in learning the free boundary, making the approach viable for complex, high-dimensional JCtrlOS problems in finance and economics beyond the aquaculture setting.

Abstract

This paper studies a joint stochastic optimal control and stopping (JCtrlOS) problem motivated by aquaculture operations, where the objective is to maximize farm profit through an optimal feeding strategy and harvesting time under stochastic price dynamics. We introduce a simplified aquaculture model capturing essential biological and economic features, distinguishing between biologically optimal and economically optimal feeding strategies. The problem is formulated as a Hamilton-Jacobi-Bellman variational inequality and corresponding free boundary problem. We develop two numerical solution approaches: First, a finite difference scheme that serves as a benchmark, and second, a Physics-Informed Neural Network (PINN)-based method, combined with a deep optimal stopping (DeepOS) algorithm to improve stopping time accuracy. Numerical experiments demonstrate that while finite differences perform well in medium-dimensional settings, the PINN approach achieves comparable accuracy and is more scalable to higher dimensions where grid-based methods become infeasible. The results confirm that jointly optimizing feeding and harvesting decisions outperforms strategies that neglect either control or stopping.

Paper Structure

This paper contains 12 sections, 30 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The Model Framework
  3. HJB Variational Inequality and Free Boundary Problem
  4. Numerical Experiments
  5. Benchmarks
  6. Finite Difference Solver
  7. Deep Solver
  8. Conclusion
  9. Appendix
  10. Exponential Feeding Rate
  11. Logistic Feeding Rate
  12. Sinusoidal Feeding Rate

Figures (3)

  • Figure 1: Comparison of the optimal control and stopping strategy obtained by the finite difference solver. The plots represent one trajectory of the simulated farm values.
  • Figure 2: Comparison of the optimal control and stopping strategy obtained by the deep learning solvers and finite difference solver. The plots represent one trajectory of the simulated farm values.
  • Figure 3: Comparison of the optimal control and stopping strategy obtained by the deep learning solvers and finite difference solver for sinusoidal feeding rate. The plots represent one trajectory of the simulated farm values.