Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

We consider regular variation of a Lévy process X := (Xt)t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

Suppose that (pn) is an infinitely divisible distribution on the non-negative integers having Lévy measure (vn). In this paper we derive a necessary and sufficient condition for the existence of the limit limn→∞ pn/vn. We also derive some other results on the asymptotic behaviour of the sequence (Pn) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt} of an infinite dam whose cumulative input {At} is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = xα and {At} is stable with index β ∊ (0, 1). In general if EAt, < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt} as t → ∞. For {At} stable with index β and r(x) = xα, α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt} in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.