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.

Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0

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.

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.

In this paper we study the tail probability of discounted aggregate claims in a continuous-time renewal model. For the case that the common claim-size 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 inter-arrival 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 continuous-time renewal model. Journal of Applied Probability 44 (2), 285–294].

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 non-standard feature of being statedependent. In addition, in order to verify the efficiency, we apply recently developed techniques based on Lyapunov inequalities. 1

In this note, we derive the optimal utility-maximizing asset allocation between a risky and risk-free asset within a variable annuity (VA) contract, which is a US-based 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

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

For a bivariate Lévy process (ξt, ηt)t≥0 and initial value V0 define the Generalised Ornstein-Uhlenbeck (GOU) process Vt: = e ξt

For a bivariate Lévy process (ξt,ηt)t≥0 and initial value V0 define the Generalised Ornstein-Uhlenbeck (GOU) process Vt: = e ξt t