Convexity and smoothness of scale functions and de Finetti's control problem

Under appropriate conditions, we obtain smoothness and convexity properties of $q$-scale functions for spectrally negative L\'evy processes. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators. As an application of the latter results to scale functions, we are able to continue the very recent work of \cite{APP2007} and \cite{Loe}. We strengthen their collective conclusions by showing, amongst other results, that whenever the L\'evy measure has a density which is log convex then for $q>0$ the scale function $W^{(q)}$ is convex on some half line $(a^*,\infty)$ where $a^*$ is the largest value at which $W^{(q)\prime}$ attains its global minimum. As a consequence we deduce that de Finetti's classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height $a^*$

Factorization of the transition matrix for the general Jacobi system

The Jacobi system on a full-line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is done by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schr\"odinger equation on the full line.Comment: 18 page

Special, conjugate and complete scale functions for spectrally negative L\'evy processes

Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative L\'evy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative L\'evy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed

Conditioning subordinators embedded in Markov processes

The running infimum of a Levy process relative to its point of issue is know to have the same range that of the negative of a certain subordinator. Conditioning a Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned to remain in a strip.Comment: 24 page

Solution of the inverse spectral problem for a convolution integro-differential operator with Robin boundary conditions

The operator of double differentiation on a finite interval with Robin boundary conditions perturbed by the composition of a Volterra convolution operator and the differentiation one is considered. We study the inverse problem of recovering the convolution kernel along with a coefficient of the boundary conditions from the spectrum. We prove the uniqueness theorem and that the standard asymptotics is a necessary and sufficient condition for an arbitrary sequence of complex numbers to be the spectrum of such an operator. A constructive procedure for solving the inverse problem is given.Comment: 10 page

Deep factorisation of the stable process II; potentials and applications

Here we propose a different perspective of the deep factorisation in Kyprianou (2015) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu transform. Here our factorisation is completely independent from the derivation in Kyprianou (2015) , moreover there is no clear way to invert the factors in Kyprianou (2015) to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form. In the spirit of the interplay between the classical Wiener-Hopf factorisation and fluctuation theory of the underlying Levy process, our analysis will produce a collection of of new results for stable processes. We give an identity for the point of closest reach to the origin for a stable process with index $\alpha\in (0,1)$ as well as and identity for the point of furthest reach before absorption at the origin for a stable process with index $\alpha\in (1,2)$. Moreover, we show how the deep factorisation allows us to compute explicitly the stationary distribution of stable processes multiplicatively reflected in such a way that it remains in the strip [-1,1].Comment: 24 pages, 8 figure

The strong matrix Stieltjes moment problem

In this paper we study the strong matrix Stieltjes moment problem. We obtain necessary and sufficient conditions for its solvability. An analytic description of all solutions of the moment problem is derived. Necessary and sufficient conditions for the determinateness of the moment problem are given.Comment: 21 page

The theory of scale functions for spectrally negative Le vy processes

The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy-Khintchine formula and its relationship to the Levy-Ito decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Le\'vy processes; Bertoin (1996), Sato (1999), Applebaum (2004), Kyprianou (2006) and Doney (2007).Comment: 92 page

Stable L\'evy processes in a cone

Ba\~nuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic tail distribution of the first time a stable (L\'evy) process in dimension $d\geq 2$ exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure of the Markov additive process that drives such processes. As a consequence of our interrogation of the underlying MAP, we are able to provide an answer by example to the open question: If the modulator of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a stationary distribution? We show how the two forms of conditioning are dual to one another. Moreover, we construct the recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes. In the spirit of several very recent works, the results presented here show that many previously unknown results of stable processes, which have long since been understood for Brownian motion, or are easily proved for Brownian motion, become accessible by appealing to the notion of the stable process as a self-similar Markov process, in addition to its special status as a L\'evy processes with a semi-tractable potential analysis

Conditioned real self-similar Markov processes

In recent work, Chaumont et al. [9] showed that is possible to condition a stable process with index ${\alpha} \in (1,2)$ to avoid the origin. Specifically, they describe a new Markov process which is the Doob h-transform of a stable process and which arises from a limiting procedure in which the stable process is conditioned to have avoided the origin at later and later times. A stable process is a particular example of a real self-similar Markov process (rssMp) and we develop the idea of such conditionings further to the class of rssMp. Under appropriate conditions, we show that the specific case of conditioning to avoid the origin corresponds to a classical Cram\'er-Esscher-type transform to the Markov Additive Process (MAP) that underlies the Lamperti-Kiu representation of a rssMp. In the same spirit, we show that the notion of conditioning a rssMp to continuously absorb at the origin also fits the same mathematical framework. In particular, we characterise the stable process conditioned to continuously absorb at the origin when ${\alpha} \in(0,1)$. Our results also complement related work for positive self-similar Markov processes in [10]