Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

Abstract

Consider a general bivariate Lévy-driven risk model. The surplus process Y , starting with Y 0 = x > 0, evolves according to dY t = Y t− dR t − dP t for t > 0, where P and R are two independent Lévy processes representing, respectively, a loss process in a world without economic factors and a process describing return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x → ∞, which confirms Paulsen's conjecture.

Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims

“…The first main result investigates the asymptotic behavior of the finite-time ruin probability in the bivariate Lévy-driven risk model, which extends the corresponding one in [14].…”

“…As for the bivariate Lévy-driven risk model, [10] gave a wealth of examples showing the exact distribution or asymptotic tail probability of Z ∞ defined in (3). An interesting paper [14] derived the asymptotics for the finite-time and infinite-time ruin probabilities in a bivariate Lévy-driven risk model with extendly regularly varying tailed jumps.…”

“…Motivated by [14], in the present paper, we aim to investigate the asymptotic behavior for the finite-time ruin probabilities in the bivariate Lévy-driven risk model with consistently varying tailed jumps. Our two other results for the finite-time and infinite-time ruin probabilities are based on a single Lévy-driven risk model, different from the one with risky investment, in which the loss process P t is still a Lévy process, whereas R t reduces to a deterministic linear function (corresponding to a constant rate of compound interest).…”

Bayesian Adjustment for Insurance Misrepresentation in Heavy-Tailed Loss Regression

“…In real applications, however, loss data often exhibit heavy-tailed features that cannot be captured by the exponential family of distributions under GLMs. As a result, statistical and probability theories for the heavy-tail phenomena have become popular research topics in the past decade (see, e.g., Hao and Tang (2012); Qi (2010); Yang et al (2018)). For statistical inference purposes, earlier papers such as Scollnik (2001Scollnik ( 2002Scollnik ( 2015 proposed to use Bayesian inference based on Markov chain Monte Carlo (MCMC) simulations, whereas books and papers including Peng and Qi (2017); Qi (2010) promoted the use of frequentist counter-parts.…”

Interplay of insurance and financial risks in a discrete-time model with strongly regular variation

“…Paulsen [41] proposed a general continuous-time risk model in which the cash flow of premiums less claims is described as a semimartingale and the log price of the investment portfolio as another semimartingale. Since then the study of ruin in the presence of risky investments has experienced a vital development in modern risk theory; some recent works include Paulsen [42], Klüppelberg and Kostadinova [33], Heyde and Wang [26], Hult and Lindskog [28], Bankovsky et al [1], and Hao and Tang [25]. During this research, much attention has been paid to an important special case of Paulsen's setup, the so-called bivariate Lévy-driven risk model, in which the two semimartingales are independent Lévy processes fulfilling certain conditions so that insurance claims dominate financial uncertainties.…”