[Paper Review] Simultaneous Ruin Probability for Two-Dimensional Brownian and Lévy Risk Models

Krzysztof Dȩbicki, Enkelejd Hashorva+1 more|arXiv (Cornell University)|Nov 11, 2018

Probability and Risk ModelsDecision Sciences15 references3 citations

TL;DR

This paper develops asymptotic approximations for simultaneous ruin probability and ruin time in a two-dimensional Brownian risk model with dependent Brownian motions, focusing on large initial capital. Using self-similarity and extreme value theory, it derives explicit formulas for the tail behavior when the initial capital tends to infinity, particularly for the case of perfect correlation (ρ = 1), enabling tractable solutions for linearly dependent spectrally positive Lévy processes such as Brownian and gamma processes.

ABSTRACT

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite-time horizon. This is not the case for the simultaneous ruin probability in two-dimensional Brownian risk model. Resorting on asymptotic theory, we derive in this contribution approximations of both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity. Given the interest in proportional reinsurance, we consider in some details the case where the correlation is 1. This model is tractable allowing for explicit formulas for the simultaneous ruin probability for linearly dependent spectrally positive Lévy processes. Examples include perturbed Brownian and gamma Lévy processes.

Motivation & Objective

• To analyze the simultaneous ruin probability in a two-dimensional Brownian risk model where two portfolios are correlated via Brownian motion.
• To derive asymptotic approximations for the simultaneous ruin probability as the initial capital u tends to infinity.
• To extend the analysis to the simultaneous ruin time and provide approximations for its tail behavior.
• To investigate the tractable case of perfect correlation (ρ = 1), where explicit formulas are obtainable for linearly dependent spectrally positive Lévy processes.
• To provide a theoretical foundation for proportional reinsurance models by analyzing joint ruin under dependence.

Proposed method

• Utilizes self-similarity of Brownian motion to reduce the time horizon to [0,1] and normalize parameters.
• Applies extreme value theory and asymptotic analysis to approximate the joint tail probability of two dependent Brownian motions exceeding linear boundaries.
• Employs a transformation based on time-changed Brownian motion and the reflection principle to derive bounds and approximations.
• Uses the Laplace transform method and integral representations to compute the asymptotic behavior of ruin probabilities.
• Derives explicit formulas for the case ρ = 1 by reducing the bivariate problem to a univariate one via linear dependence.
• Applies results from Gaussian process extremes, including Gordon's inequality and the theory of first-passage times, to handle dependent processes.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the simultaneous ruin probability in a two-dimensional Brownian risk model as the initial capital u tends to infinity?
• RQ2 How does the correlation ρ between the two portfolios affect the joint ruin probability, particularly in the case ρ = 1?
• RQ3 Can explicit formulas be derived for the simultaneous ruin probability when the two risk processes are linearly dependent and spectrally positive?
• RQ4 What is the asymptotic distribution of the simultaneous ruin time in the two-dimensional model?
• RQ5 How do the premium rates c₁ and c₂ and initial capital levels influence the joint ruin probability in the large-deviation regime?

Key findings

• For ρ = 1, the simultaneous ruin probability admits an explicit formula, reducing the bivariate problem to a univariate one via linear dependence.
• When a < ρ, the asymptotic approximation of the simultaneous ruin probability is given by 2√(2π(1−ρ²)) exp((c₂−ρc₁)²/(2(1−ρ²))), which is derived from a Laplace transform integral.
• For a = ρ, the asymptotic approximation involves the standard normal CDF: 2√(2π(1−ρ²)) exp((c₂−ρc₁)²/(2(1−ρ²))) Φ((c₁ρ−c₂)/√(1−ρ²)).
• The simultaneous ruin time τ_sim(u) satisfies P(u²(1−τ_sim(u)) > x | τ_sim(u) ≤ 1) → 1 as u → ∞, indicating that ruin occurs near time 1 with high probability.
• The upper and lower bounds on the ruin probability are derived via self-similarity and time-scaling, leading to tight asymptotic control.
• The results are extended to linearly dependent spectrally positive Lévy processes, including Brownian and gamma processes, under the ρ = 1 assumption.