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Reinforcement learning for irreversible reinsurance problems: the randomized singular control approach

Zongxia Liang, Xiaodong Luo, Xiang Yu

TL;DR

The paper tackles learning optimal irreversible reinsurance strategies in a continuous-time setting by formulating a two-stage entropy-regularized, randomized singular-control problem. It derives a time-consistent equilibrium for the outer problem and provides an explicit inner-solution structure, enabling a two-stage actor-critic RL framework that jointly learns inner value functions and outer activation boundaries. Policy evaluation and policy iteration are developed to update the inner singular-control law and the outer boundary, with convergence supported by analytical results in a solvable model. Numerical experiments show that randomized exploration improves convergence and parameter recovery compared to a non-randomized baseline, offering a robust approach for continuous-time RL in singular control problems.

Abstract

This paper studies the continuous-time reinforcement learning for stochastic singular control with the application to an infinite-horizon irreversible reinsurance problems. The singular control is equivalently characterized as a pair of regions of time and the augmented states, called the singular control law. To encourage the exploration in the learning procedure, we propose a randomization method for the singular control laws, new to the literature, by considering an auxiliary singular control and entropy regularization. The exploratory singular control problem is formulated as a two-stage optimal control problem, where the time-inconsistency issue arises in the outer problem. In the specific model setup with known model coefficients, we provide the full characterization of the time-consistent equilibrium singular controls for the two-stage problem. Taking advantage of the solution structure, we can consider the proper parameterization of the randomized equilibrium policy and the value function when the model is unknown and further devise the actor-critic reinforcement learning algorithms. In the numerical experiment, we present the superior convergence of parameter iterations towards the true values based on the randomized equilibrium policy and illustrate how the exploration may advance the learning performance in the context of singular controls.

Reinforcement learning for irreversible reinsurance problems: the randomized singular control approach

TL;DR

The paper tackles learning optimal irreversible reinsurance strategies in a continuous-time setting by formulating a two-stage entropy-regularized, randomized singular-control problem. It derives a time-consistent equilibrium for the outer problem and provides an explicit inner-solution structure, enabling a two-stage actor-critic RL framework that jointly learns inner value functions and outer activation boundaries. Policy evaluation and policy iteration are developed to update the inner singular-control law and the outer boundary, with convergence supported by analytical results in a solvable model. Numerical experiments show that randomized exploration improves convergence and parameter recovery compared to a non-randomized baseline, offering a robust approach for continuous-time RL in singular control problems.

Abstract

This paper studies the continuous-time reinforcement learning for stochastic singular control with the application to an infinite-horizon irreversible reinsurance problems. The singular control is equivalently characterized as a pair of regions of time and the augmented states, called the singular control law. To encourage the exploration in the learning procedure, we propose a randomization method for the singular control laws, new to the literature, by considering an auxiliary singular control and entropy regularization. The exploratory singular control problem is formulated as a two-stage optimal control problem, where the time-inconsistency issue arises in the outer problem. In the specific model setup with known model coefficients, we provide the full characterization of the time-consistent equilibrium singular controls for the two-stage problem. Taking advantage of the solution structure, we can consider the proper parameterization of the randomized equilibrium policy and the value function when the model is unknown and further devise the actor-critic reinforcement learning algorithms. In the numerical experiment, we present the superior convergence of parameter iterations towards the true values based on the randomized equilibrium policy and illustrate how the exploration may advance the learning performance in the context of singular controls.

Paper Structure

This paper contains 8 sections, 6 theorems, 135 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation and Entropy Regularization for Randomized Singular Control Laws
  3. Optimal Singular Control Laws for the Inner Problem
  4. Time-Consistent Equilibrium Singular Control for the Outer Problem
  5. Actor-Critic RL using the Two-Stage Formulation
  6. Policy evaluation
  7. Policy iteration
  8. RL Algorithms and Numerical Implementations

Key Result

Lemma 3.1

For any $C^{2}(\mathbb{R})$ function $\tilde{F}(y)$ , we have the expansion at $q\downarrow 0$ that

Figures (2)

  • Figure 1: Comparison of learned parameters $\theta_{i}, i=1,2,3$ of non-randomized Algorithm \ref{['benchmark0']} and randomized Algorithm \ref{['rand0']} versus the true parameters during episodes of training.
  • Figure 2: Left panel: comparison of learned optimal boundary $\bar{x}$ of non-randomized Algorithm \ref{['benchmark0']} and randomized Algorithm \ref{['rand0']} versus the true optimal boundary during episodes of training. Right panel: comparison of $L^{\infty}$ error of learned value function $\Phi(x;\bar{x})$ of non-randomized Algorithm \ref{['benchmark0']} and randomized Algorithm \ref{['rand0']} during episodes of training.

Theorems & Definitions (25)

  • Remark 1
  • Definition 2.1: Activation time of auxiliary singular control
  • Remark 2
  • Remark 3
  • Lemma 3.1
  • proof
  • Definition 4.1: Admissible auxiliary singular control law
  • Remark 4
  • Definition 4.2: Equilibrium auxiliary singular control law
  • Theorem 4.3: Verification theorem of the outer problem
  • ...and 15 more