We consider an insurance risk model for the cashflow of an insurance company, which invests its reser.v.e into a portfolio consisting of risky and riskless assets. The price of the risky asset is modeled by an exponential Lévy process. We derive the integrated risk process and the corresponding discounted net loss process. We calculate certain quantities as characteristic functions and moments. We also show under weak conditions stationarity of the discounted net loss process and derive the left and right tail behaviour of the model. Our results show that the model carries a high risk, which may originate either from large insurance claims or from the risky investment.

We consider a stochastic model for the wealth of an insurance company which has the possibility to invest into a risky and a riskless asset under a constant mix strategy. The total claim amount is modeled by a compound Poisson process and the price of the risky asset follows a general exponential Levy process. We investigate the resulting integrated risk process and the corresponding discounted net loss process.

We consider a positive stationary generalized Ornstein-Uhlenbeck process Vt = e −ξt ( ∫ t e ξs− dηs + V0 for t ≥ 0, 0 ∫ k k−1 and the increments of the integrated generalized Ornstein-Uhlenbeck process Ik = Vt − dLt, k ∈ N, where (ξt,ηt,Lt)t≥0 is a three-dimensional Lévy process independent of the starting random variable V0. The genOU model is a continuous time version of a stochastic recurrence equation. Hence, our models include, in particular, continuous time versions of ARCH(1) and GARCH(1,1) processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of (Vt)t≥0 and (Ik)k∈N. Furthermore, we present a central limit result for (Ik)k∈N. Regular variation and point process convergence play a crucial role in establishing the statistics of (Vt)t≥0 and (Ik)k∈N. The theory can be applied to the COGARCH(1,1) and the Nelson diffusion model.

In this paper we present a review on the extremal behavior of stationary continuous-time processes with emphasis on generalized Ornstein-Uhlenbeck processes. We restrict our attention to heavy-tailed models like heavy-tailed Ornstein-Uhlenbeck processes or continuous-time GARCH processes. The survey includes the tail behavior of the stationary distribution, the tail behavior of the sample maximum and the asymptotic behavior of sample maxima of our models. 1

Properties of the law of the integral R 1 0 c Nt dYt are studied, where c> 1 and f(Nt; Yt); t 0g is a bivariate Levy process such that fNtg and fYtg are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalised Ornstein-Uhlenbeck process. The law is either continuous-singular or absolutely continuous, and sufficient conditions for each case are given. Under the condition of independence of fNtg and fYtg, it is shown that is continuous-singular if b=a is sufficiently small for xed c, or if c is su ciently large for fixed a and b, or if c is in the set of Pisot-Vijayaraghavan numbers, which includes all integers bigger than 1, for any a and b, and that, for Lebesgue almost every c, is absolutely continuous if b=a is sufficiently large. The law is infinitely divisible if fNtg and fYtg are independent, but not in general. Complete characterisation of infinite divisibility is given for and for the symmetrisation of. Under the condition that is in nitely divisible, the continuity properties of the convolution t power of are also studied. Some results are extended to the case where fYtg is an integer valued Levy process with finite second moment.

Large insurance losses happen infrequently, but they happen. In this paper we present the standard distribution models used in fire, wind–storm or flood insurance. We also present the classical Cramér-Lundberg model for the total claim amount and some more recent extensions. The classical insurance risk measure is the ruin probability and we give a full account of the ruin event in such models. Finally, we present some results for an integrated insurance risk model, where also investment risk is taken into account.

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We compare the largest jump in a Lévy process Xt up till time t, i.e, Yt = sup{|Xs − Xs− | : s ≤ t}, to the two-sided maximal value of the process, Let Mt = sup{|Xs | : s ≤ t}. T(r) = inf{t> 0: |Xt |> r}, r> 0, be the two-sided passage time out of the two-sided strip [−r, r]. Then we show that Yt is negligible with respect to Mt for small times, i.e., lim t↓0

We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent Lévy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an independent fractional Brownian motion (FBM) as integrator. We show that the GFOU process is well-defined by checking the existence of the integral included in the process, and investigate its properties. It is proved that the process has a stationary version and exhibits long memory. We also find that the process satisfies a certain stochastic differential equation. Our underlying intention is to introduce long memory into the GOU process which has short memory without losing the possibility of jumps. Note that both FOU and GOU processes have found application in a variety of fields as useful alternatives to the Ornstein-Uhlenbeck (OU) process.

Let (ξ, η) be a bivariate Lévy process such that the integral ∫ ∞ 0 e−ξt − dηt converges almost surely. We characterise, in terms of their Lévy measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I: = ∫ ∞ 0 g(ξt) dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ ∞ 0 g(ξt) dYt, where Y is an almost surely strictly increasing stochastic process, independent of ξ. 1