Ruin probabilities with a Markov chain interest 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.

In this paper, we study the ruin problem with investment in a general framework where the business part X is a Levy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a Levy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R.

In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distribution -- Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.

A revisit to ruin probabilities in the presence of heavy-tailed insurance and financial risks

Uniform estimate on finite time ruin probabilities with random interest rate

Calculation of finite time ruin probabilities for some risk models

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.

In the paper we study the finite-time ruin probability in a general risk model with constant interest force, in which the claim sizes are pairwise quasi-asymptotically independent and arrive according to an arbitrary counting process, and the premium process is a general stochastic process. For the case that the claim-size distribution belongs to the consistent variation class, we obtain an asymptotic formula for the finite-time ruin probability, which holds uniformly for all time horizons varying in a relevant infinite interval. The obtained result also includes an asymptotic formula for the infinite-time ruin probability.

Padé approximants for finite time ruin probabilities

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

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.

In this paper, we study ruin probabilities in two generalized risk models. The effects of timing of payments and interest on the ruin probabilities in the models are considered. The rates of interest are assumed to have a dependent autoregressive structure. Generalized Lundberg inequalities for the ruin probabilities are derived by a renewal recursive technique. An illustrative application is given to the compound binomial risk process.

This paper studies a bidimensional renewal risk model with constant force of interest and heavy-tailed claims. We extend the existing models through allowing arbitrary dependences between each pair of inter-arrival times of the two kinds of insurance claims. A precise asymptotic formula for the finite-time ruin probability is obtained when the claims have subexponential tails. Further restricting the claim-size distributions within the class of extended regular variation, we derive the corresponding asymptotic formula for the infinite-time ruin probability.

Opening session.- Invited address.- Invited lecture: Some major issues in economics and insurance developments.- Main lectures.- Risk convolution calculations.- Risk sharing, incentives and moral hazard.- State-dependent utility, the demand for insurance and the value of safety.- Separation of risk parameters.- Weighted Markov processes with an application to risk theory.- Practical models in credibility theory, including parameter estimation.- Rate making and the society's sense of fairness.- The impact of reinsurance on the insurer's risk.- Chains of reinsurance.- Net stop-loss ordering and related ordering.- Limit theorems for risk processes.- Semi-Markov models in economics and insurance.- Loss distributions: estimation, large sample theory, and applications.- Rating of non proportional reinsurance treaties based on ordered claims.- Resistant line fitting in actuarial science.- Quantitative models of pension costs.- Credibility: estimation of structural parameters.- Short communications.- Population and social security projections for Bangladesh.- Stability of premium principles under maximum entropy perturbations.- Practical rating of variable accident excess-loss premiums.- Motor premium rating.- The mean square error of a randomly discounted sequence of uncertain payments.- Operational time: a short and simple existence proof.- Simulation in actuarial work. Some computational problems.- Bayesian sequential analysis of multivariate point processes.- The actuary in practice.- The influence of reinsurance limits on infinite time ruin probabilities.- Some Berry-Esseen theorems for risk processes.- Some notes on the methods of calculation of life assurance premiums in the United Kingdom.- A stochastic model for investment variables in the United Kingdom.- Inflationary effects on pension plans: wage and benefit patterns.