Results 1  10
of
13
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 ..."
Abstract

Cited by 98 (9 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
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 ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
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.
Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE
 Insurance Math. Econom
, 2004
"... We consider the problem of maximizing the expected utility of discounted dividend payments of an insurance company. The risk process, describing the insurance business of the company, is modeled as Brownian motion with drift. We mainly consider power utility and special emphasis is given to the limi ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We consider the problem of maximizing the expected utility of discounted dividend payments of an insurance company. The risk process, describing the insurance business of the company, is modeled as Brownian motion with drift. We mainly consider power utility and special emphasis is given to the limiting behavior when the coefficient of risk aversion tends to zero. We then find convergence of the corresponding dividend strategies to the classical case of maximizing the expected dividend payments.
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. The l ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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].
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
IMPORTANCE SAMPLING OF COMPOUNDING PROCESSES
"... Compounding processes, also known as perpetuities, play an important role in many applications; in particular, in time series analysis and mathematical finance. Apart from some special cases, the distribution of a perpetuity is hard to compute, and large deviations estimates sometimes involve compli ..."
Abstract
 Add to MetaCart
Compounding processes, also known as perpetuities, play an important role in many applications; in particular, in time series analysis and mathematical finance. Apart from some special cases, the distribution of a perpetuity is hard to compute, and large deviations estimates sometimes involve complicated constants which depend on the complete distribution. Motivated by this, we propose provably efficient importance sampling algorithms which apply to qualitatively different cases, leading to light and heavy tails. Both algorithms have the nonstandard feature of being statedependent. In addition, in order to verify the efficiency, we apply recently developed techniques based on Lyapunov inequalities. 1
Optimal asset allocation in life annuities: a note
, 2001
"... In this note, we derive the optimal utilitymaximizing asset allocation between a risky and riskfree asset within a variable annuity (VA) contract, which is a USbased savings and decumulation investment product. We are interested in the interaction between financial risk, mortality risk and consum ..."
Abstract
 Add to MetaCart
In this note, we derive the optimal utilitymaximizing asset allocation between a risky and riskfree asset within a variable annuity (VA) contract, which is a USbased savings and decumulation investment product. We are interested in the interaction between financial risk, mortality risk and consumption, towards the end of the life cycle. Our main result is that for constant relative risk aversion (CRRA) preferences and geometric Brownian motion (GBM) dynamics, the optimal asset allocation during the annuity decumulation (payout) phase is identical to the accumulation (savings) phase, which is the classical Merton
On Continuity Properties of the Law of Integrals of Lévy Processes
"... 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 ..."
Abstract
 Add to MetaCart
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