The Exact Asymptotics for Hitting Probability of a Remote Orthant by a Multivariate Lévy Process: The Cramér Case

Abstract

For a multivariate Lévy process satisfying the Cramér moment condition and having a drift vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by the multivariate ruin problem introduced in F. Avram et al. (2008) in the two-dimensional case. Our solution relies on the analysis from Y. Pan and K. Borovkov (2017) for multivariate random walks and an appropriate time discretization.

Simultaneous ruin probability for two-dimensional brownian risk model

“…The leading term in the asymptotics of the finite time horizon ruin probability is with Note that if , then . In the infinite time horizon, the leading term in the asymptotic of simultaneous ruin probability equals for some positive ; see [5, 7]. Consequently, in the bivariate BRM the same relation as (2) holds between the simultaneous ruin probability in the finite and infinite time horizon.…”

“…Related results for the infinite time horizon are obtained in [2, 3, 5, 7, 16, 18, 22]. The first three papers consider the case that .…”

Pandemic-type Failures in Multivariate Brownian Risk Models

“…There are no results in the literature investigating ψ k (S, T, u) for general k. The particular case k = d, for which ψ d (S, T, u) coincides with the simultaneous ruin probability has been studies in different contexts, see e.g., [3][4][5][6][7][8][9][10][11][12][13][14]. The case d = 2 of Brm has been recently investigated in [2].…”

Pandemic-type failures in multivariate Brownian risk models

“…There are no results in the literature investigating ψ k (S, T , u) for general k. The particular case k = d, for which ψ d (S, T , u) coincides with the simultaneous ruin probability has been studies in different contexts, see e.g., Lieshout and Mandjes (2007), Avram et al (2008a), Avram et al (2008b), Dȩbicki et al (2018), Ji and Robert (2018), Foss et al (2017), Pan and Borovkov (2019), Borovkov and Palmowski (2019), Ji (2020), Hu and Jiang (2013), Samorodnitsky and Sun (2016), and Dombry and Rabehasaina (2017). The case d = 2 of Brm has been recently investigated in Dȩbicki et al (2020).…”