On the Application of Laplace Transform to the Ruin Problem with Random Insurance Payments and Investments in a Risky Asset

Paper: arXiv 2508.07235

Authors: Viktor Antipov

Abstract

This paper considers the ruin problem with random premiums, whose densities have rational Laplace transforms, and investments in a risky asset whose price follows a geometric Brownian motion. The asymptotic behavior of the ruin probability for large initial capital values is investigated.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical derivation using integro-differential equations, Laplace transforms, and the Frobenius method to establish asymptotic ruin probabilities under a general jump distribution model. However, it is a purely mathematical probability paper with no empirical data, empirical validation, or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Ruin Probability Asymptotics with Random Premiums & Risky Assets] --> B[Inputs: GBM Risky Asset & Premiums with Rational Laplace Transforms]
    B --> C[Methodology: Apply Laplace Transform to Integro-Differential Equations]
    C --> D[Computational Process: Derive Asymptotic Behavior for Large Initial Capital]
    D --> E[Outcome: Explicit Asymptotic Formula for Ruin Probability]