Parisian ruin probability for two-dimensional Brownian risk model

TL;DR

This paper derives exact asymptotics for the probability of non-simultaneous Parisian ruin in a bivariate Brownian risk model, considering large initial capital and short time-in-red durations, relevant for financial risk analysis.

Contribution

It provides the first precise asymptotic formulas for non-simultaneous Parisian ruin probabilities in a correlated bivariate Brownian setting.

Findings

Exact asymptotics for ruin probabilities as initial capital grows large.

Asymptotic behavior characterized for short time-in-red proportional to 1/u^2.

Applicable to prolonged ruin analysis in financial risk models.

Abstract

Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $\rho \in (-1,1).$ Parisian ruin is defined as a classical ruin that happens over an extended period of time, the so-called time-in-red. We derive exact asymptotics for the non-simultaneous Parisian ruin of the company conditioned on the event of non-simultaneous ruin happening. We are interested in finding asymptotics of such problem as $u \to \infty$ and with the length of time-in-red being of order $\frac{1}{u^2},$ where $u$ represents initial capital for the companies. Approximation of this problem is of interest for the analysis of Parisian ruin probability in bivariate Brownian risk model, which is a standard way of defining prolonged ruin models in the financial markets.