Asymptotic behaviour of the finite‐time ruin probability in renewal risk models

Abstract

In this paper we study the tail behaviour of the probability of ruin within finite time t, as initial risk reserve x tends to infinity, for the renewal risk model with strongly subexponential claim sizes. The asymptotic formula holds uniformly for t ∈[ f (x), ∞), where f (x) is an infinitely increasing function, and substantially extends the result of Tang (Stoch. Models 2004; 20:281-297) obtained for the class of claim distributions with consistently varying tails. Two examples illustrate the result.

The exponential moment tail of inhomogeneous renewal process

“…having a finite expectation (see Lemma 3.5). Leipus and Šiaulys (2009) considered the asymptotic behavior of finite-time ruin probability in the renewal risk model…”

Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk

“…The author of the paper uses the assertion of Theorem 1 for function ϕ(x) = (1 + x) q with q > 0 to get the main term of the asymptotics for the probability ψ(u, T ). Leipus and Šiaulys considered the asymptotic behavior of ψ(u, T ) in [22,23] but for subexponentially distributed r.v. 's {Z 1 , Z 2 , .…”

“….}. In their proofs, the assertion of Theorem 1 was used for function ϕ(x) = exp(ρx) with some ρ > 0 (see Lemma 3.3 in [22] and Lemma 2.1 in [23]). In the case of exponential function, Theorem 1 implies the following assertion.…”

Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims

“…Tang [6] also provided a rather complete list of references on the study of the finite-time ruin probability. See Leipus and Šiaulys [12,13] for further extensions of this result.…”