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

Paper: arXiv 2510.02910

Authors: Kevin Kamm

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematics including Hamilton-Jacobi-Bellman variational inequalities, free boundary problems, and high-dimensional numerical schemes (finite difference and PINNs). It includes a code repository and data, demonstrating empirical implementation, though the model is a simplified ’toy’ aquaculture application rather than a full backtest on real financial data.

Research Flowchart

  flowchart TD
  A["Research Goal: Find optimal feeding & harvesting strategy<br>maximizing aquaculture profit under uncertainty"] --> B["Develop Aquaculture Model<br>- Stochastic price dynamics<br>- Biological growth process<br>- Cost & revenue functions"]

  B --> C["Formulate JCtrlOS Problem<br>- HJB Variational Inequality<br>- Free Boundary Problem<br>- Control (Feeding) & Stopping (Harvest)"]

  C --> D["Numerical Solution Methods"]

  D --> E["Finite Difference Scheme<br>- Grid-based solver<br>- Benchmark for accuracy"]
  D --> F["Physics-Informed Neural Network<br>- PINN + Deep Optimal Stopping<br>- Scalable to higher dimensions"]

  E & F --> G["Computational Experiments<br>- Compare JCtrlOS vs. Suboptimal Strategies<br>- Vary dimensions & parameters"]

  G --> H["Key Findings & Outcomes"]
  H --> H1["Joint optimization outperforms<br>single or no-control strategies"]
  H --> H2["Finite differences excel in<br>medium-dimensional settings"]
  H --> H3["PINN approach achieves<br>comparable accuracy & scalability"]

Cited References (9)

1. On the impact of biological risk in aquaculture valuation and decision making (2025) — link
2. On the impact of feeding cost risk in aquaculture valuation and decision making — → 2309.02970
3. Approximation Schemes for Mixed Optimal Stopping and Control Problems with Nonlinear Expectations and Jumps (2019) — link
4. A Weak Dynamic Programming Principle for Combined Optimal Stopping/Stochastic Control with $\mathcal E ^ f $-expectations (2016) — link
5. DGM: A deep learning algorithm for solving partial differential equations (2018) — link
6. Numerical Methods for Stochastic Control Problems in Continuous Time (2001) — link
7. Neural optimal stopping boundary (2025) — link
8. Deep optimal stopping (2020) — link

1 of 9 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.