Results 1  10
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15
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 98 (10 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 34 (4 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
Levy Integrals and the Stationarity of generalised OrnsteinUhlenbeck processes
"... The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t ..."
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Cited by 21 (9 self)
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The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t
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 14 (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 13 (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.
Ruin models with investment income
 Probab. Surv
"... This paper is a survey of recent progress in the theory of ruin for risk processes that earn investment return on invested assets. ..."
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Cited by 4 (0 self)
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This paper is a survey of recent progress in the theory of ruin for risk processes that earn investment return on invested assets.
Extremes of ContinuousTime Processes
"... In this paper we present a review on the extremal behavior of stationary continuoustime processes with emphasis on generalized OrnsteinUhlenbeck processes. We restrict our attention to heavytailed models like heavytailed OrnsteinUhlenbeck processes or continuoustime GARCH processes. The survey ..."
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Cited by 2 (0 self)
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In this paper we present a review on the extremal behavior of stationary continuoustime processes with emphasis on generalized OrnsteinUhlenbeck processes. We restrict our attention to heavytailed models like heavytailed OrnsteinUhlenbeck processes or continuoustime 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
A uniform asymptotic estimate for discounted aggregate claims with subexponential tails
 Insurance Math. Econom
"... In this paper we study the tail probability of discounted aggregate claims in a continuoustime renewal model. For the case that the common claimsize distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with som ..."
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Cited by 1 (0 self)
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In this paper we study the tail probability of discounted aggregate claims in a continuoustime renewal model. For the case that the common claimsize distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and interarrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuoustime renewal model. Journal of Applied Probability 44 (2), 285–294].
Uniform Tail Asymptotics for the Stochastic Present Value of Aggregate Claims in the Renewal Risk Model
, 2009
"... Consider an insurer who is allowed to make riskfree and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claimsize distribution is of Pareto ..."
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Cited by 1 (1 self)
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Consider an insurer who is allowed to make riskfree and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claimsize distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same asymptotic formula holds for the finitetime and infinitetime ruin probabilities. Restricting our attention to the socalled constant investment strategy, we show how the insurer adjusts his investment portfolio to maximize the expected terminal wealth subject to a constraint on the ruin probability.
Distributional properties of exponential functionals of Lévy processes ∗
"... 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 1 (1 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.