Asymptotics for a discounted systemic risk measure in a multi-dimensional risk model with dependent claim sizes and stochastic return

Abstract

Consider a multi-dimensional renewal risk model, in which an insurer simultaneously operates more than one line of business, and is allowed to make both risk-free and risky investments. The claim sizes from all business lines are triggered by a series of common shocks, whose arrival times constitute a renewal counting process. The price process of the investment portfolio is described as a geometric Lévy process. In this study, we consider two cases that the claim sizes from different business lines are asymptotically dependent or asymptotically independent. Under the framework of regular variation, we study the asymptotic behavior of a discounted systemic risk measure, which is proposed by Li (2022) to describe the instant expected shortfall of the insurer at the moment when one of its business lines suffers crisis. The obtained results indicate that the asymptotics for the discounted systemic risk measure depend on neither the price process of the investment nor the renewal shock-number process, and, in addition, the asymptotic independence also gives no contribution to the asymptotic formula.