On the Total Duration of Negative Surplus of a Risk Process with Two-step Premium Function

We consider a risk reserve process whose premium rate reduces from cd to cu when the reserve comes above some critical value v. In the model of Cramer-Lundberg with initial capital u ≥ 0, we obtain the probability that ruin does not occur before the first up-crossing of level v. When u \u3c v, following H. Gerber and E. Shiu (1997), we derive the probability that starting with initial capital u ruin occurs and the severity of ruin is not bigger than v. Further we express the probability of ruin in the two step premium function model - ψ (u,v), by the last two probabilities. Our assumptions imply that the surplus process will go to infinity almost surely. This entails that the process will stay below zero only temporarily. We derive the distribution of the total duration of negative surplus and obtain its Laplace transform and mean value. As a consequence of these results, under certain conditions in the Model of Cramer-Lundberg we obtain the expected value of the severity of ruin. In the end of the paper we give examples with exponential claim sizes

Functional Transfer Theorems for Maxima of Stationary Processes

2000 Mathematics Subject Classification: 60G70, 60F12, 60G10.In this paper we discuss the problem of finding the limit process of sequences of continuous time random processes, which are constructed as properly affine transformed maxima of random number identically distributed random variables. The max-increments of these processes are dependent. First we work under the well known conditions D (un) and D' (un) of Leadbetter, Lindgren and Rootzen, (1983). Further we investigate the case of moving average sequence. The distribution function of the noise components is assumed to have regularly varying tails or is subexponential and belongs to the max-domain of attraction of Gumbel distribution or belongs to the max-domain of attraction of Weibull distribution. We work with random time-components which are a.s. strictly increasing to infinity. In particular their counting process is a mixed Poisson process or a renewal process with regularly varying tails with parameter β ∈ (0, 1). Here is proved that such sequences of random processes converges weakly to a compound extremal process.This research is partially supported by the NFSI, Grant VU-MI-105/2005 of the Ministry of Science and Education, Bulgaria

(G, λ)-Extremal Processes and Their Relationship with Max-Stable Processes

2000 Mathematics Subject Classification: 60G70, 60G18.The study of G-extremal processes was initiated by S. Resnick and M. Rubinovich (1973). Here we transform these processes by a non-decreasing and right-continuous function λ : [0, ∞) → [0, ∞) and investigate relationship between (G; λ)-extremal processes and max-stable processes. We prove that for the processes with independent max-increments if one of the following three statements is given, the other two are equivalent: a) Y is a max-stable process; b) Y is a (G; λ)-extremal process; c) Y is a self-similar extremal process.This paper is partially supported by NFSI-Bulgaria, Grant No VU-MI-105/2005

Merging of bivariate compound Binomial processes with shocks

The paper investigates a discrete time Binomial risk model with different types of polices and shock events may influence some of the claim sizes. It is shown that this model can be considered as a particular case of the classical compound Binomial model. As far as we work with parallel Binomial counting processes in infinite time, if we consider them as independent, the probability of the event they to have at least once simultaneous jumps would be equal to one. We overcome this problem by using thinning instead of convolution operation. The bivariate claim counting processes are expressed in two different ways. The characteristics of the total claim amount processes are derived. The risk reserve process and the probabilities of ruin are discussed. The deficit at ruin is thoroughly investigated when the initial capital is zero. Its mean, probability mass function and probability generating function are obtained. We show that although the probability generating function of the global maxima of the random walk is uniquely determined via its probability mass function and vice versa, any compound geometric distribution with non-negative summands has uncountably many stochastically equivalent compound geometric presentations. The probability to survive in much more general settings, than those, discussed here, for example in the Anderson risk model, has uncountably many Beekman's convolution series presentations.Comment: submitte