R. A. Doney

Overshoots and undershoots of Lèvy processes

We obtain a new fluctuation identity for a general Levy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Kluppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766–1801] concerning asymptotic overshoot distribution of a particular class of Levy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Levy process is spectrally one sided

Fluctuation Theory for Lévy Processes

Recently there has been renewed interest in fluctuation theory for Levy processes. Inthis brief survey we describe several aspects of this topic, including Wiener-Hopf factorisation,the ladder processes, Spitzer’s condition, the asymptotic behaviour of Levy processes at zero and infinity, and other path properties. Some open problems are also presented.

Cramer's estimate for Lévy processes

It is shown that the usual method of establishing Cramers estimate also works for Levy processes.

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process Xt as t↓0 or t→∞, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t↓0; for example, we may have Xt/t →P+∞ (t↓0) (weak divergence to +∞), whereas Xt/t→∞ a.s. (t↓0) is impossible (both are possible when t→∞), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. “Almost sure stability” of Xt, i.e., Xt tending to a nonzero constant a.s. as t→∞ or as t↓0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to “strong law” behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.

Spitzer's condition and ladder variables in random walks

SummarySpitzers condition holds for a random walk if the probabilities ρn=P{n > 0} converge in Cèsaro mean to ϱ, where 0<ϱ<1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that ρn converges to ϱ. This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.

Some Excursion Calculations for Spectrally One-sided Lévy Processes

1. Introduction 2. Preliminaries 3. Exit results for the reflected processes References

Spitzer's condition for random walks and Lévy Processes

Spitzers condition holds for a random walk S if the probabilities ρn=P{Sn> 0} converge in Cesaro mean to ρ, and for a Levy process X at ∞ (at 0, respectively) if t1 ∫0t ρ(s)ds→ ρ as t→ ∞(0), where ρ(s)=P{Xs >0}. It has been shown in Doney [4] that if 0 < ρ < 1 then this happens for a random walk if and only if ρn converges to ρ. We show here that this result extends to the cases ρ = 0 and ρ = 1, and also that Spitzers condition holds for a Levy process at ∞(0) if and only if ρ(t) → ρ as t → ∞(0).

Wiener-Hopf factorization revisited and some applications

A reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights

THE ASYMPTOTIC BEHAVIOR OF DENSITIES RELATED TO THE SUPREMUM OF A STABLE PROCESS

P(S1 > x) v A� −1 x −� as x ! 1 and P(S1 � x) v B� −1 � −1 x �� as x # 0. [Here � = P(X1 > 0) and A and B are known constants.] It is also known that S1 has a continuous density, m say. The main point of this note is to show that m(x) v Ax −(�+1) as x ! 1 and m(x) v Bx ��−1 as x # 0. Similar results are obtained for related densities.

Stochastic bounds for Lévy processes

Using the Wiener-Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Levy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Levy process versions of many known results about the large-time behavior of random walks. This is illustrated by establishing a comprehensive theorem about Levy processes which converge to ∞ in probability.