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Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks

Zan Yu, Lianzeng Zhang

TL;DR

Addressing the computation of the Gerber-Shiu function $\Phi_b(u)$ under constant interest and a dividend barrier, the paper applies physics-informed neural networks (PINNs) to solve the associated integro-differential equations. PINNs embed the differential equation residuals into a loss and use automatic differentiation to train a mesh-free surrogate for $\Phi_b(u)$. Its contributions include a general PINN-based framework for barrier and barrier-free settings, demonstrated across Exponential, Erlang(2), and mixture-of-exponentials claim sizes, achieving near $10^{-5}$ relative error against collocation methods. It highlights the practical flexibility of PINNs for ruin-related quantities and suggests directions to optimize integral discretization and boundary handling for large values of $u$.

Abstract

In this paper, we propose a new efficient method for calculating the Gerber-Shiu discounted penalty function. Generally, the Gerber-Shiu function usually satisfies a class of integro-differential equation. We introduce the physics-informed neural networks (PINN) which embed a differential equation into the loss of the neural network using automatic differentiation. In addition, PINN is more free to set boundary conditions and does not rely on the determination of the initial value. This gives us an idea to calculate more general Gerber-Shiu functions. Numerical examples are provided to illustrate the very good performance of our approximation.

Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks

TL;DR

Addressing the computation of the Gerber-Shiu function under constant interest and a dividend barrier, the paper applies physics-informed neural networks (PINNs) to solve the associated integro-differential equations. PINNs embed the differential equation residuals into a loss and use automatic differentiation to train a mesh-free surrogate for . Its contributions include a general PINN-based framework for barrier and barrier-free settings, demonstrated across Exponential, Erlang(2), and mixture-of-exponentials claim sizes, achieving near relative error against collocation methods. It highlights the practical flexibility of PINNs for ruin-related quantities and suggests directions to optimize integral discretization and boundary handling for large values of .

Abstract

In this paper, we propose a new efficient method for calculating the Gerber-Shiu discounted penalty function. Generally, the Gerber-Shiu function usually satisfies a class of integro-differential equation. We introduce the physics-informed neural networks (PINN) which embed a differential equation into the loss of the neural network using automatic differentiation. In addition, PINN is more free to set boundary conditions and does not rely on the determination of the initial value. This gives us an idea to calculate more general Gerber-Shiu functions. Numerical examples are provided to illustrate the very good performance of our approximation.
Paper Structure (9 sections, 40 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 9 sections, 40 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries on risk model and Gerber-Shiu function
  3. Methodology
  4. Deep neural networks
  5. Physics-Informed Neural Networks (PINN)
  6. Numerical illustrations
  7. Concluding remarks
  8. A representation of the discounted penalty function
  9. The exact solution of $\Phi_{\infty}(0)$

Figures (5)

  • Figure 1: Visualization of a deep neural network. In this example, the number of layers $L$ is $4$. In other words, the network consists of an input layer, three hidden layers, and an output layer.
  • Figure 2: Computing the Gerber-Shiu functions with Exponential claim size density where red lines denote the value curves using collocation method and green lines denote estimate value curves. (a) ruin probability; (b) Laplace transform of ruin time; (c) expected claim size causing ruin; (d) expected deficit at ruin.
  • Figure 3: Computing the Gerber-Shiu functions of Erlang (2) claim sizes density by mean value curves. (a) ruin probability; (b) Laplace transform of ruin time; (c) expected claim size causing ruin; (d) expected deficit at ruin.
  • Figure 4: Computing the Gerber-Shiu functions of combination-of-exponentials claim sizes density by mean value curves. (a) ruin probability; (b) Laplace transform of ruin time; (c) expected claim size causing ruin; (d) expected deficit at ruin
  • Figure 5: Computing the Gerber-Shiu functions with a constant dividend barrier. (a) exponential claim size density; (b) Erlang (2) claim size density; (c) combination of exponentials

Theorems & Definitions (1)

  • Remark 1