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

Abstract

Let {ξ 1 , ξ 2 , . . .} be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability P(sup n 0 n i=1 ξ i > x) can be bounded above by 1 exp{− 2 x} with some positive constants 1 and 2 . A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.

The Exponential Estimate of the Ultimate Ruin Probability for the Non-Homogeneous Renewal Risk Model

“…These assumptions would allow analysis of the model in more realistic cases of insurance. The main results of this paper extend and complement the results of other authors, who have considered the exponential estimate of the ruin probability in the non-homogeneous renewal risk model (see Andrulytė et al 2015;Kievinaitė and Šiaulys 2018;Castañer et al 2013;Grandell and Schmidli 2011;Liu et al 2017a).…”

“…Some authors consider models in which claim amounts are divided in several lines by supposing some dependence relations between these lines (see Fu and Ng 2017;Guo et al 2017;Yang and Yuen 2016). In this work, we consider a non-homogeneous renewal risk model with independent, but not necessarily identically distributed claims and inter-arrival times like in articles by Andrulytė et al (2015), Kievinaitė and Šiaulys (2018) and Rȃducan et al (2015).…”

“…It is neither clear whether the requirement plays the role of the net profit condition in the non-homogeneous renewal risk model, nor whether it is the basis to get exponential upper bound for the probability of ruin. In articles by Andrulytė et al (2015) and Kievinaitė and Šiaulys (2018), it is supposed that the "net profit condition" holds on average, i.e., 1 n…”

Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model

“…In such a case, we can use also the upper estimate of ruin probability, which usually decreases with increasing initial capital. The useful estimates for the nonhomogeneous models we can find in [27][28][29][30][31] among others. For instance, results of [28,29] imply that ψ(u) c 1 exp{−c 2 u}, u 0, for all above examples with a positive constants c 1 , and c 2 depending on the numerical characteristics of the random claims X, Y and Z, generating a three-risk discrete time model.…”

“…The useful estimates for the nonhomogeneous models we can find in [27][28][29][30][31] among others. For instance, results of [28,29] imply that ψ(u) c 1 exp{−c 2 u}, u 0, for all above examples with a positive constants c 1 , and c 2 depending on the numerical characteristics of the random claims X, Y and Z, generating a three-risk discrete time model. On the other hand, the proven statements ignite the hope that similar algorithms can be found to calculate values of the ultimate time survival probability for the general multi-risk discrete time model.…”