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26
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 119 (12 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 41 (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 23 (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
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 17 (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.
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 15 (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.
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 7 (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.
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 3 (1 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].
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 3 (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
On the decomposition of the ruin probability for a jumpdiffusion surplus process compounded by a geometric Brownian motion
 North American Actuarial Journal
, 2006
"... Assume that the surplus of an insurer follows a jumpdiffusion process and the insurer would invest its surplus in a risky asset, whose prices are modelled by a geometric Brownian motion. The resulting surplus for the insurer is called as a jumpdiffusion surplus process compounded by the geometric Br ..."
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Cited by 2 (0 self)
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Assume that the surplus of an insurer follows a jumpdiffusion process and the insurer would invest its surplus in a risky asset, whose prices are modelled by a geometric Brownian motion. The resulting surplus for the insurer is called as a jumpdiffusion surplus process compounded by the geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or by oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities, one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integrodifferential equations, and it is shown that all the three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jumpdiffusion surplus process is an exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not exponential functions, when claim sizes are exponentially distributed.
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 2 (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.