Finite Time Ruin Probabilities and Large Deviations for Generalized Compound Binomial Risk Models

Calculation of finite time ruin probabilities for some risk models

Research on Ruin Problems with Two-type by-claims and Delay Period Based on Compound Binomial Risk Model

Kluppelberg and Stadtmuller (1998) proved a precise asymptotic formula for the ruin probability of the classical interest force and regularly varying tailed claims when the initial capital u tends to infinity . This paper extends their results in several aspects as follows: First, the risk models are the Conditional Poisson and Non-Qici Poisson Process respectively; Second, the ruin probability is replaced by the finite time ruin probability within time T and the claimsize is of Subexponential family which is more wide then that of regularly varying's family. At last, the diffusion term of Brownian Motion is considered. construct et ψ(u;T) be the finite time ruin probability in the renewal risk model, where u is the initial capital of the company and $T$ denotes some given time bound. Under the assumption that the distribution of the claim size belongs to the Extended regular variation class, this paper obtains an asymptotic formula for ψ(u;T). This result improves the related works in the recent literature.

Due to having useful properties in approximating to the other distributions and mathematically tractable, phase type distributions are commonly used in actuarial risk theory. Claim occurrence time and individual claim size distributions are modelled by phase type distributions in literature. This paper aims to calculate the survival probabilities of an insurance company under the assumption that compound binomial risk model where the individual claim sizes are distributed as discrete Phase Type distribution

Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes

In this paper; survival (non-ruin) probability after a definite time period of an insurance company is studied in a discrete time model based on nonhomogenous claim occurrences. Furthermore, distributions of the minimum and maximum levels of surplus in compound binomial risk model with nonhomogeneous claim occurrences are obtained and some of its characteristics are given.

In this paper, we revisit the finite-time ruin probability in the classical compound Poisson risk model. Traditional general solutions to finite-time ruin problems are usually expressed in terms of infinite sums involving the convolutions related to the claim size distribution and their integrals, which can typically be evaluated only in special cases where the claims follow exponential or (more generally) mixed Erlang distribution. We propose to tackle the partial integro-differential equation satisfied by the finite-time ruin probability and develop a new approach to obtain a solution in terms of bivariate Laguerre series as a function of the initial surplus level and the time horizon for a large class of light-tailed claim distributions. To illustrate the versatility and accuracy of our proposed method which is easy to implement, numerical examples are provided for claim amount distributions such as generalized inverse Gaussian, Weibull and truncated normal where closed-form convolutions are not available in the literature.

We find the optimal dividend strategy in two related risk models under an affine penalty for ruin. The first risk model is the classical Cramer–Lundberg risk model, and the second is the so-called dual risk model. Under both models, for exponentially distributed jumps, we show that the optimal dividend strategy is a barrier strategy and obtain the barrier explicitly. Moreover, we prove that the optimal barrier increases with respect to the parameters of the affine penalty, while the penalized value function decreases with respect to the penalty. We also compute the expected time until ruin and show that the expected time of ruin increases with respect to the parameters of the affine penalty. Finally, we present some numerical examples to demonstrate the relationship between the results for the classical and dual risk models. Our main contributions are in comparing the classical and dual risk models side-by-side and in obtaining explicit expressions for the penalized value functions, the optimal barriers, and the expected times of ruin. Also, we contrast the free-boundary condition associated with the barrier strategies in the two models.

Padé approximants for finite time ruin probabilities

The finite time ruin probability in actuarial science and in companies throughout their operating lives remains a major and very complex concern for the latter. Given the complexity of the occurrence of natural risk phenomena, the mathematical determination of this probability can prove particularly complex. This mathematical complexity represents a considerable obstacle for researchers in this field. Previous research has mainly focused on risk models constructed from Markovian processes. In this paper, we address the modeling risk where the number of claims is approximated by a variable memory counting process based on a self-excited Hawkes process. We subsequently considered that the claim amounts are independent random variables and identically distributed following an exponential law. This allowed us to determine an expression for the probability of ruin in finite time. We proposed a simulation to visualize the evolution of the probability of ruin as a function of the initial reserve and time.

Uniform estimate on finite time ruin probabilities with random interest rate

In this paper, we consider the credit risk model with dependent rates of interest. The rates of interest {\( {I_n},n = 1,2, \)…} are assumed to have an autoregressive structure, we obtained the integral equations for the penalty function by using the analysis methods in probability. Furthermore, the finite time ruin probability, the joint distribution of surplus immediately before ruin and the deficit at ruin are also obtained.

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

Ruin probabilities with a Markov chain interest model

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.