[Paper Review] On matrix exponential approximations of the infimum of a spectrally negative Levy process

Florin Avram, András Horváth+1 more|arXiv (Cornell University)|Oct 9, 2012

Random Matrices and Applications22 references3 citations

TL;DR

This paper develops high-order matrix-exponential approximations for the infimum of spectrally negative Lévy processes, leveraging moment fitting and Padé approximation techniques to improve accuracy in first-passage and ruin probability calculations. It introduces a new approximation for the perturbed Cramér-Lundberg model and extends Johnson-Taaffe approximations to fit arbitrarily high moments, significantly improving upon classical methods like Renyi, De Vylder, and Whitt-Ramsay.

ABSTRACT

We recall four open problems concerning constructing high-order matrix-exponential approximations for the infimum of a spectrally negative Levy process (with applications to first-passage/ruin probabilities, the waiting time distribution in the M/G/1 queue, pricing of barrier options, etc). On the way, we provide a new approximation, for the perturbed Cramer-Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taaffe, which fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whitt-Ramsay. Obtaining such approximations which fit the Laplace transform at infinity as well would be quite useful.

Motivation & Objective

• To address the lack of high-order moment-based approximations for the infimum of spectrally negative Lévy processes in first-passage and ruin problems.
• To develop a new matrix-exponential approximation for the perturbed Cramér-Lundberg model that ensures admissibility and high moment fitting.
• To generalize Johnson and Taaffe’s moment-fitting approximations to arbitrary order, surpassing current methods like Renyi, De Vylder, and Whitt-Ramsay.
• To investigate the feasibility of constructing approximations that match both low-order moments and the asymptotic behavior of the Laplace transform at infinity.
• To ensure admissibility of matrix-exponential approximations by verifying nonnegativity of coefficients and proper phase-type representation.

Proposed method

• Utilizes moment-based fitting of the Laplace transform of the infimum distribution using matrix-exponential functions.
• Applies Padé and two-point Padé approximations to the Laplace transform to achieve high-order moment fitting.
• Employs symbolic inversion of the Pollaczek-Khinchine formula to derive exact ruin probabilities for benchmarking.
• Introduces a new approximation method for the perturbed Cramér-Lundberg model that preserves asymptotic behavior and admissibility.
• Extends Johnson-Taaffe approximations to allow fitting of arbitrarily many moments via rational approximation of the Laplace transform.
• Validates approximations using numerical tables of exact and approximate ruin probabilities under Gamma and mixed exponential claim distributions.

Experimental results

Research questions

• RQ1 What are the fundamental challenges in constructing high-order moment-based approximations for the infimum of spectrally negative Lévy processes?
• RQ2 Can matrix-exponential approximations be constructed to fit more than three moments while ensuring admissibility (nonnegative densities)?
• RQ3 How can approximations be designed to match both low-order moments and the asymptotic behavior of the Laplace transform at infinity?
• RQ4 To what extent do the proposed approximations outperform classical methods like Renyi, De Vylder, and Whitt-Ramsay in terms of accuracy and moment fitting?
• RQ5 Can Johnson-Taaffe-type approximations be generalized to arbitrary order while preserving moment fitting and admissibility?

Key findings

• The proposed matrix-exponential approximation for the perturbed Cramér-Lundberg model yields valid, admissible ruin probabilities even when direct moment-based Padé approximations fail.
• For Gamma claims with α=2.5, β=1, the new method achieves relative errors below 0.003% in the intermediate regime, significantly outperforming Renyi and De Vylder approximations.
• Theorem 3 and Theorem 5 approximations reduce relative errors to less than 0.002% for x ≥ 1.5 in the Gamma(2.5,1) case, demonstrating high accuracy.
• For mixed exponential claims, the approximation from Theorem 1 consistently outperforms the one from Remark 12, especially at low σ=0.1, with relative errors below 0.005% in most regions.
• The Johnson-Taaffe family of approximations is generalized to arbitrary order, enabling fitting of any number of moments while maintaining admissibility and convergence.
• Numerical results show that the new approximations maintain high accuracy across different claim distributions and perturbation levels, with error decreasing as σ increases.