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31
On Continuity Properties of the Law of Integrals of Lévy Processes
, 2008
"... 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 f ..."
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Cited by 30 (4 self)
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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.
Integrated insurance risk models with exponential Lévy investment
 Insurance Math. Econ
, 2008
"... 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 disc ..."
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Cited by 19 (4 self)
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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.
Fluctuation theory and exit systems for positive selfsimilar Markov processes
 Preprint. AND ASYMPTOTIC nTUPLE LAWS AT FIRST AND LAST PASSAGE 563
, 2009
"... For a positive selfsimilar Markov process, X, we construct a local time for the random set, �, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) ..."
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Cited by 15 (4 self)
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For a positive selfsimilar Markov process, X, we construct a local time for the random set, �, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive selfsimilar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set � and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Lévy process associated to X via Lamperti’s transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finitedimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012–1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates. 1. Introduction. In
CONTINUITY PROPERTIES AND INFINITE DIVISIBILITY OF STATIONARY DISTRIBUTIONS OF SOME GENERALISED ORNSTEINUHLENBECK PROCESSES
"... 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 OrnsteinUhlenbeck process. Th ..."
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Cited by 12 (1 self)
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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 OrnsteinUhlenbeck process. The law is either continuoussingular 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 continuoussingular 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 PisotVijayaraghavan 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.
Optimal Investment for Insurers, When the Stock Price Follows an Exponential Lévy Process
 Insurance Math. Econ
, 2005
"... 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 ..."
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Cited by 9 (1 self)
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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.
Asymptotic results for sample autocovariance functions and extremes of integrated generalized OrnsteinUhlenbeck processes
, 2008
"... We consider a positive stationary generalized OrnsteinUhlenbeck process Vt = e −ξt ( ∫ t e ξs− dηs + V0 for t ≥ 0, 0 ∫ k k−1 and the increments of the integrated generalized OrnsteinUhlenbeck process Ik = Vt − dLt, k ∈ N, where (ξt,ηt,Lt)t≥0 is a threedimensional Lévy process independent of the s ..."
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Cited by 9 (1 self)
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We consider a positive stationary generalized OrnsteinUhlenbeck process Vt = e −ξt ( ∫ t e ξs− dηs + V0 for t ≥ 0, 0 ∫ k k−1 and the increments of the integrated generalized OrnsteinUhlenbeck process Ik = Vt − dLt, k ∈ N, where (ξt,ηt,Lt)t≥0 is a threedimensional 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.
Distributional properties of exponential functionals of Lévy processes
, 2012
"... We study the distribution of the exponential functional I(ξ, η) = ∫ ∞ 0 exp(ξt−)dηt, where ξ and η are independent Lévy processes. In the general setting, using the theory of Markov processes and Schwartz distributions, we prove that the law of this exponential functional satisfies an integral equ ..."
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Cited by 7 (2 self)
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We study the distribution of the exponential functional I(ξ, η) = ∫ ∞ 0 exp(ξt−)dηt, where ξ and η are independent Lévy processes. In the general setting, using the theory of Markov processes and Schwartz distributions, we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in [9]. In the special case when η is a Brownian motion with drift, we show that this integral equation leads to an important functional equation for the Mellin transform of I(ξ, η), which proves to be a very useful tool for studying the distributional properties of this random variable. For general Lévy process ξ (η being Brownian motion with drift) we prove that the exponential functional has a smooth density on R \ {0}, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that ξ has some positive exponential moments we establish an asymptotic behaviour of P(I(ξ, η)> x) as x → +∞, and under similar assumptions on the negative exponential moments of ξ we obtain a precise asymptotic expansion of the density of I(ξ, η) as x → 0. Under further assumptions on the Lévy process ξ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when ξ has hyperexponential jumps.
Exact conditions for no ruin for the generalised OrnsteinUhlenbeck process
, 2008
"... For a bivariate Lévy process (ξt,ηt)t≥0 the generalised OrnsteinUhlenbeck (GOU) process is defined as Vt: = e ξt z + ∫ t 0 e −ξs− dηs, t ≥ 0, where z ∈ R. We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are st ..."
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Cited by 6 (2 self)
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For a bivariate Lévy process (ξt,ηt)t≥0 the generalised OrnsteinUhlenbeck (GOU) process is defined as Vt: = e ξt z + ∫ t 0 e −ξs− dηs, t ≥ 0, where z ∈ R. We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the Lévy process and reveal the effect of the dependence relationship between ξ and η. We also present technical results which explain the structure of the lower bound of the GOU.