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Stochastic Control Barrier Functions for Economics

David van Wijk

TL;DR

This work extends safety-critical control to economics by applying Control Barrier Functions (CBFs) and Stochastic Control Barrier Functions (SCBFs) to three problems: deterministic advertising, stochastic advertising, and portfolio optimization. It integrates an Optimal Primary Controller (via PMP or HJB) with Active Set Invariance Filter (ASIF) or SCBF-based constraints to enforce state safety like market-share caps and wealth floors, including corrections to SCBF conditions based on recent findings. The approach is validated through numerical simulations—both single trials and Monte Carlo studies—demonstrating that barriers effectively constrain unsafe trajectories while preserving near-optimal performance and exhibiting low computation times. The results highlight the practical potential of safety-aware decision-making in finance under uncertainty, with future work pointing toward higher-dimensional assets and slack-variable formulations to improve scalability.

Abstract

Control barrier functions (CBFs) and safety-critical control have seen a rapid increase in popularity in recent years, predominantly applied to systems in aerospace, robotics and neural network controllers. Control barrier functions can provide a computationally efficient method to monitor arbitrary primary controllers and enforce state constraints to ensure overall system safety. One area that has yet to take advantage of the benefits offered by CBFs is the field of finance and economics. This manuscript re-introduces three applications of traditional control to economics, and develops and implements CBFs for such problems. We consider the problem of optimal advertising for the deterministic and stochastic case and Merton's portfolio optimization problem. Numerical simulations are used to demonstrate the effectiveness of using traditional control solutions in tandem with CBFs and stochastic CBFs to solve such problems in the presence of state constraints.

Stochastic Control Barrier Functions for Economics

TL;DR

This work extends safety-critical control to economics by applying Control Barrier Functions (CBFs) and Stochastic Control Barrier Functions (SCBFs) to three problems: deterministic advertising, stochastic advertising, and portfolio optimization. It integrates an Optimal Primary Controller (via PMP or HJB) with Active Set Invariance Filter (ASIF) or SCBF-based constraints to enforce state safety like market-share caps and wealth floors, including corrections to SCBF conditions based on recent findings. The approach is validated through numerical simulations—both single trials and Monte Carlo studies—demonstrating that barriers effectively constrain unsafe trajectories while preserving near-optimal performance and exhibiting low computation times. The results highlight the practical potential of safety-aware decision-making in finance under uncertainty, with future work pointing toward higher-dimensional assets and slack-variable formulations to improve scalability.

Abstract

Control barrier functions (CBFs) and safety-critical control have seen a rapid increase in popularity in recent years, predominantly applied to systems in aerospace, robotics and neural network controllers. Control barrier functions can provide a computationally efficient method to monitor arbitrary primary controllers and enforce state constraints to ensure overall system safety. One area that has yet to take advantage of the benefits offered by CBFs is the field of finance and economics. This manuscript re-introduces three applications of traditional control to economics, and develops and implements CBFs for such problems. We consider the problem of optimal advertising for the deterministic and stochastic case and Merton's portfolio optimization problem. Numerical simulations are used to demonstrate the effectiveness of using traditional control solutions in tandem with CBFs and stochastic CBFs to solve such problems in the presence of state constraints.
Paper Structure (24 sections, 1 theorem, 43 equations, 10 figures)

This paper contains 24 sections, 1 theorem, 43 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Functions
  4. Active Set Invariance Filter
  5. Stochastic Control Barrier Functions
  6. Problem Formulation
  7. Optimal Advertising
  8. Primary Controller
  9. Barrier Constraints
  10. Stochastic Optimal Advertising
  11. Primary Controller
  12. Stochastic Barrier Constraints
  13. Portfolio Optimization
  14. Primary Controller
  15. Stochastic Barrier Constraints
  16. ...and 9 more sections

Key Result

Theorem 1

Suppose there exists a function $h : \mathcal{X} \rightarrow \mathbb{R}$ and a extended class $\mathcal{K}$ function $\alpha$ where, for all $x \in \mathcal{X}$ satisfying $h(\boldsymbol{x}) > 0$, there exists $\boldsymbol{u}\in \mathcal{U}$ such that Then, for all $t \ge 0$, $\texttt{Pr}\{\boldsymbol{x}(t) \in C_S\}=1$, provided that $\boldsymbol{x}(0)\in \rm Int(\mathcal{C}_S)$.

Figures (10)

  • Figure 1: Visualization of safety condition ensuring forward invariance of $\mathcal{C}_S$ without strengthening function.
  • Figure 2: Market share over time demonstrating compliance with anti-trust regulations for deterministic advertising.
  • Figure 3: Control inputs over time demonstrating safety interventions for deterministic advertising.
  • Figure 4: Single trial market share over time demonstrating compliance with anti-trust regulations for stochastic advertising.
  • Figure 5: Single trial control inputs over time demonstrating safety interventions for stochastic advertising.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1