Tail Probabilities for a Risk Process with Subexponential Jumps in a Regenerative and Dwfusion Environment

In this paper we fmd a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reflected Brownian motion, square functions of the Omstein-Uhlenbeck process and splitting reflected Brownian bridges. In particular, we consider a non-Markovian intensity process. This paper is concerned with a risk theory subject to a combination of two features: a stochastic modulation and regularly varying claim size distributions. We consider a canonical surplus process {S (t), t 2 0) given by S(t) = C u it , i = 1 where (N (t) , t 2 0) is a Cox process with an underlying cadlhg process (X (t) , t 2 0). That is, if a realization of the process (X (t) , t 2 0) is x (t) E 3 [0, + a) , then for a nonnegative measurable function A: R-, R , u (0) the process (N (t), t 2 0) has the same law as a nonhomogeneous Poisson process (N'x)(t), t 2 0) with an intensity function q t) = A(x(t)). The process { A (X (t)), t 3 0) is called an intensity process. Thus stochastic modulation means that the surplus process is not time-homogeneous, but evolves in some random environment. A detailed discussion of Cox processes and their impact on risk theory is to be found in Grandell [I91 and Rolski et al. [33]. The claim sizes independent of the process {N(t), t 2 0) with a common distribution function Fo(x). Let u be an initial reserve and assume that S (t) +-a a.e. as t + +a. An inftnite horizon retin probability is then $ (u) = P (sup S (t) > u). The model, in which {N (t), t 2 0) is Coxian, is called the Bjurk-Grandell model which goes to the pioneering paper Bjtirk and Grandell [lo]. In that paper one derives by a martingale approach an exponential upper bound of $ (u) when an intensity process has piecewise constant realizations and claim sizes are light-tailed. Further generalizations can be found in Embrechts et al. [16] (finite time non-Markovian intensities) and Grigelionis [22]. In applications one needs also to consider environment which changes more widely like it is in the case of diffusions or Gaussian processes. Grandell and Schmidli [21] …