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- R. A. Doney, A. E. Kyprianou
- 2004

We obtain a new fluctuation identity for a general Lévy 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 Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab.… (More)

Let X be either the branching diffusion corresponding to the operator Lu + β(u2 − u) on D ⊆ Rd [where β(x) ≥ 0 and β ≡ 0 is bounded from above] or the superprocess corresponding to the operator Lu + βu − αu2 on D ⊆ Rd (with α > 0 and β is bounded from above but no restriction on its sign). Let λc denote the generalized principal eigenvalue for the operator… (More)

- J D Biggins, A E Kyprianou
- 1996

In the discrete-time supercritical branching random walk there is a Kesten-Stigum type result for the martingales formed by the Laplace transform of the nth generation positions. Roughly, this says that for suitable values of the argument of the Laplace transform the martingales converge in mean provided an X log X condition holds. Here it is established… (More)

- F Avram, A E Kyprianou, M R Pistorius
- 2007

We consider spectrally negative L evy process and determine the joint Laplace transform of the exit time and exit position from an interval containing the origin of the process reeected in its supremum. In the literature of uid models, this stopping time can be identiied as the time to buuer-overrow. The Laplace transform is determined in terms of the scale… (More)

- A. E. Kyprianou
- 2004

Recently Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503], using probabilistic methods alone, has given new proofs for the existence, asymptotics and uniqueness of travelling wave solutions to the K-P-P equation. Following in this vein we outline alternative probabilistic proofs. Specifically the techniques are confined to the study of additive and… (More)

We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally,… (More)

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give… (More)

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, Random Fields and Applications (2011) 119–146, Electron. J.… (More)

- A. E. Kyprianou
- 2008

Following recent developments in Hubalek and Kyprianou [28] the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Lévy processes which are completely explicit. This will follow as a consequence of an observation in the aforementioned paper which permits feeding the theory of… (More)

- Andreas E. Kyprianou, Budhi Arta Surya
- Finance and Stochastics
- 2007

We revisit the previous work of Leland [13], Leland and Toft [12] and Hilberink and Rogers [8] on optimal capital structure and show that the issue of an optimal endogenous default level can be dealt with analytically and numerically when the underlying source of randomness is replaced by that of a general spectrally negative Lévy process. By working with… (More)