The exact asymptotics for hitting probability of a remote orthant by a multivariate L\'evy process: the Cram\'er case

TL;DR

This paper derives precise asymptotic probabilities for a multivariate Lévy process to hit a distant orthant, extending previous work on multivariate ruin problems and employing advanced probabilistic analysis techniques.

Contribution

It provides the first exact asymptotic results for the hitting probability of a remote orthant by a multivariate Lévy process under the Cramér condition, generalizing prior two-dimensional findings.

Findings

Exact asymptotics for hitting probabilities derived

Method extends analysis from multivariate random walks to Lévy processes

Results applicable to multivariate ruin problems and risk assessment

Abstract

For a multivariate L\'evy process satisfying the Cram\'er 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.