Parisian ruin probability for two-dimensional Brownian risk model

Abstract

Let (W1(s),W2(t)),s,t≥0 be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation ρ∈(−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→∞ and with the length of time-in-red being of order 1u2, 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.