The Laplace transform of the time of ruin in the Cramér--Lundberg model with Erlang $E_2(μ)$ claims: spectrum, exact moments and matrix reduction

Paper: arXiv 2611.01886

Authors: Goluba Yurieva

Abstract

We study the Laplace transform of the time of ruin in the classical Cramér–Lundberg model with claims having an Erlang distribution of shape parameter 2. We derive an integro-differential equation with two boundary conditions. For every positive discount rate, we show that its characteristic polynomial has one negative root and two distinct positive roots. Boundedness excludes the growing exponential term and yields a representation of the transform as a sum of two decaying exponentials. We construct a finite-dimensional matrix reduction of the convolution chain. For a model parameter set, we obtain exact expressions for the ruin probability, the mean time of ruin, and higher moments, including the third moment at zero initial surplus. We prove analyticity in the discount rate and justify the resulting moment series. We consider the analytic continuation of the representation to negative discount rates separately. For Erlang claims with an arbitrary positive integer shape parameter, we establish a closed linear system and the asymptotics of the root continuing the zero root at zero discount.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on complex mathematical derivations for the Laplace transform of ruin time. While it presents novel exact solutions and matrix reductions, it lacks any empirical validation or real-world data application. The clarity is good for the target audience, but reproducibility is limited by the absence of code or data.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Laplace Transform of Ruin Time with Erlang E2 Claims] --> B{Methodology: Integro-Differential Eq. & Matrix Reduction};
    B --> C[Inputs: Cramér--Lundberg Model, Erlang E2(μ) Claims];
    C --> D{Computational Processes: Characteristic Polynomial Roots, Matrix System Solution};
    D --> E[Key Findings 1: Transform Representation as Decaying Exponentials, Finite-Dimensional Matrix Reduction];
    E --> F[Key Findings 2: Exact Ruin Probability & Higher Moments (E.g., Mean, 3rd Moment), Analyticity in Discount Rate];
    F --> G[Key Findings 3: Closed Linear System & Root Asymptotics for General Erlang Claims (E_k)];