Last Exit before an Exponential Time for Spectrally Negative Lévy Processes
نویسنده
چکیده
Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).
منابع مشابه
On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum
Consider a spectrally one-sided Lévy process X and reflect it at its past infimum I. Call this process Y . For spectrally positive X, Avram et al. [2] found an explicit expression for the law of the first time that Y = X − I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y , if X is spectrally negative. Subsequently, we find an expression fo...
متن کاملApplying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy processes to Path Functionals such as First Passage Times, Undershoots and Overshoots
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Lévy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Lévy model) and insurance...
متن کاملParisian Ruin Probability for Spectrally Negative Lévy Processes
In this note we give, for a spectrally negative Lévy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero which length exceeds a certain fixed period r. The formula involves only the scale function of the spectrally negative Lévy process and the distribution of the process at time r.
متن کاملInfinitely Divisibility of Solutions of Some Semi-stable Integro-differential Equations and Exponential Functionals of Lévy Processes
0 (1 ∧ x2) ν(dx) < +∞, are uniquely determined by the distribution of a spectrally negative infinitely divisible random variable, with characteristic exponent ψ. L(α,ψ,γ) is known to be the infinitesimal generator of a 1 α -semi-stable Feller semigroup on R+, which has been introduced by Lamperti [18]. The functions are expressed in terms of a new family of power series which includes, for inst...
متن کاملStochastic control for spectrally negative Lévy processes
Three optimal dividend models are considered for which the underlying risk process is a spectrally negative Lévy process. The first one concerns the classical dividends problem of de Finetti for which we give sufficient conditions under which the optimal strategy is of barrier type. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves...
متن کامل