Meromorphic Levy processes and their fluctuation identities

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention in particular the many cases of spectrally negative Levy processes, hyper-exponential and generalized hyper-exponential Levy processes, Lamperti-stable processes, Hypergeometric processes, Beta-processes and Theta-processes. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener-Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Levy-Khintchin exponent, all of which appear on the imaginary axis of the complex plane. The specific structure of the M-class Wiener-Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.Comment: 12 figure

The W,ZW,Z scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends

First passage problems for spectrally negative L\'evy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" $W$ (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function $Z$ (Avram, Kyprianou and Pistorius 2004). Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "$W,Z$ alphabet" for a great variety of first passage problems. We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem. One interesting use of the kit is for recognizing relationships between apparently unrelated problems -- see Lemma 3. Last but not least, it turned out recently that once the classic $W,Z$ are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) L\'evy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative L\'evy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with Omega killing (Li and Palmowski 2017)

Probabilistic foundation of nonlocal diffusion and formulation and analysis for elliptic problems on uncertain domains

2011 Summer.Includes bibliographical references.In the first part of this dissertation, we study the nonlocal diffusion equation with so-called Lévy measure ν as the master equation for a pure-jump Lévy process. In the case ν ∈ L1(R), a relationship to fractional diffusion is established in a limit of vanishing nonlocality, which implies the convergence of a compound Poisson process to a stable process. In the case ν ∉ L1(R), the smoothing of the nonlocal operator is shown to correspond precisely to the activity of the underlying Lévy process and the variation of its sample paths. We introduce volume-constrained nonlocal diffusion equations and demonstrate that they are the master equations for Lévy processes restricted to a bounded domain. The ensuing variational formulation and conforming finite element method provide a powerful tool for studying both Lévy processes and fractional diffusion on bounded, non-simple geometries with volume constraints. In the second part of this dissertation, we consider the problem of estimating the distribution of a quantity of interest computed from the solution of an elliptic partial differential equation posed on a domain Ω(θ) ⊂ R2 with a randomly perturbed boundary, where (θ) is a random vector with given probability structure. We construct a piecewise smooth transformation from a partition of Ω(θ) to a reference domain Ω in order to avoid the complications associated with solving the problems on Ω(θ). The domain decomposition formulation is exploited by localizing the effect of the randomness to boundary elements in order to achieve a computationally efficient Monte Carlo sampling procedure. An a posteriori error analysis for the approximate distribution, which includes a deterministic error for each sample and a stochastic error from the effect of sampling, is also presented. We thus provide an efficient means to estimate the distribution of a quantity of interest via a Monte Carlo sampling procedure while also providing a posteriori error estimates for each sample

On a class of stochastic models with two-sided jumps

In this paper a stochastic process involving two-sided jumps and a continuous downward drift is studied. In the context of ruin theory, the model can be interpreted as the surplus process of a business enterprise which is subject to constant expense rate over time along with random gains and losses. On the other hand, such a stochastic process can also be viewed as a queueing system with instantaneous work removals (or negative customers). The key quantity of our interest pertaining to the above model is (a variant of) the Gerber-Shiu expected discounted penalty function (Gerber and Shiu in N. Am. Actuar. J. 2(1):48-72, 1998) from ruin theory context. With the distributions of the jump sizes and their inter-arrival times left arbitrary, the general structure of the Gerber-Shiu function is studied via an underlying ladder height structure and the use of defective renewal equations. The components involved in the defective renewal equations are explicitly identified when the upward jumps follow a combination of exponentials. Applications of the Gerber-Shiu function are illustrated in finding (i) the Laplace transforms of the time of ruin, the time of recovery and the duration of first negative surplus in the ruin context; (ii) the joint Laplace transform of the busy period and the subsequent idle period in the queueing context; and (iii) the expected total discounted reward for a continuous payment stream payable during idle periods in a queue. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 21 Feb 201

The Likelihood of Mixed Hitting Times

We present a method for computing the likelihood of a mixed hitting-time model that specifies durations as the first time a latent L\'evy process crosses a heterogeneous threshold. This likelihood is not generally known in closed form, but its Laplace transform is. Our approach to its computation relies on numerical methods for inverting Laplace transforms that exploit special properties of the first passage times of L\'evy processes. We use our method to implement a maximum likelihood estimator of the mixed hitting-time model in MATLAB. We illustrate the application of this estimator with an analysis of Kennan's (1985) strike data.Comment: 35 page