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

Abstract

In this work, the non-homogeneous risk model is considered. In such a model, claims and inter-arrival times are independent but possibly non-identically distributed. The easily verifiable conditions are found such that the ultimate ruin probability of the model satisfies the exponential estimate exp{− u} for all values of the initial surplus u 0. Algorithms to estimate the positive constant are also presented. In fact, these algorithms are the main contribution of this work. Sharpness of the derived inequalities is illustrated by several numerical examples.

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.…”

“…R * we enlarge its probability of small value comparing to r. v. R. Intuitively, this creates a less dangerous claim and leads to enlarged survival probability. More precisely, the equality (29) implies that…”

Bi-seasonal discrete time risk model with income rate two

“…Scrolling across the timeline, an observable works of Gerber and Shiu on the risk collective models could be highlighted: [5], [6], [7] and [8]. Recently, many research papers on the related risk models as in (1) are occurring per year, see for example [9], [10], [11], [12], [13], [14], [15] and references therein.…”

Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

“…That is why numerous bounds and approximations for the ruin probability are established and investigated in different risk models (see, e.g., [1][2][3][4][5] and references therein). Construction of exponential bounds is one of the key problems that is studied in risk theory (see, e.g., [46], [47] (Theorem 1), [48] (Theorem 1), [49] (Section 2), [50] (Theorem 2), [51] ([Theorem 5.1), [52] (Section 3), [53] (Theorems 2-4) and [54] (Theorem 2)). In particular, for the classical compound Poisson risk model, the exponential bound, which is also called the Lundberg inequality, can be derived in different ways (see, e.g., [1][2][3]), one of which is the martingale approach introduced by Gerber [46].…”