Special Issue "Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics"

A special issue of Risks (ISSN 2227-9091).

Deadline for manuscript submissions: 30 April 2019

Special Issue Editor

Guest Editor
Prof. Dr. Florin Avram

Laboratoire de Mathématiques Appliquées, Université de Pau, France
Website | E-Mail
Interests: stochastic processes, risk, mathematical finance, inventory, queueing and population dynamics

Special Issue Information

Dear Colleagues,

It has long been well-known that exit problems for one-dimensional Lévy processes are easier when there are jumps in one direction only. In the last few years, this intuition became more precise: We know now that a great variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two “q-harmonic functions” (or scale functions, or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property and hold in principle for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with Omega state dependent killing, certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W, Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein-Uhlenbeck or Feller branching diffusion with phase-type jumps).

Motivated by these considerations, this Special Issue aims to review and push further the state-of-the-art on the following topics:

  • W, Z formulas for exit problems of the Lévy and diffusion classes (including drawdown problems)
  • W, Z formulas for quasi-stationary distributions
  • Asymptotic results
  • Extensions to random walks, Markov additive processes, Omega models, processes with Parisian reflection or absorbtion, processes with state-dependent drift, etc.
  • Optimal stopping, dividends, real options, etc
  • Numeric computation of the scale functions

Prof. Dr. Florin Avram
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Risks is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 350 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Exit/first passage problems
  • Levy processes
  • Diffusions
  • Scale functions
  • Drawdown/trailing stop
  • State-dependent parameters
  • Excursion theory
  • Phase-type distributions
  • Hypergeometric functions
  • Optimal control

Published Papers (2 papers)

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Research

Open AccessArticle Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint
Received: 18 December 2018 / Revised: 28 January 2019 / Accepted: 29 January 2019 / Published: 31 January 2019
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Abstract
We consider the optimal bail-out dividend problem with fixed transaction cost for a Lévy risk model with a constraint on the expected present value of injected capital. To solve this problem, we first consider the optimal bail-out dividend problem with transaction cost and [...] Read more.
We consider the optimal bail-out dividend problem with fixed transaction cost for a Lévy risk model with a constraint on the expected present value of injected capital. To solve this problem, we first consider the optimal bail-out dividend problem with transaction cost and capital injection and show the optimality of reflected ( c 1 , c 2 ) -policies. We then find the optimal Lagrange multiplier, by showing that in the dual Lagrangian problem the complementary slackness conditions are met. Finally, we present some numerical examples to support our results. Full article
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Open AccessArticle Fluctuation Theory for Upwards Skip-Free Lévy Chains
Received: 20 July 2018 / Revised: 13 September 2018 / Accepted: 16 September 2018 / Published: 18 September 2018
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Abstract
A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free Lévy chains, i.e., for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of Lévy [...] Read more.
A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free Lévy chains, i.e., for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of Lévy processes—several results, however, can be made more explicit/exhaustive in the compound Poisson setting. Importantly, the scale functions admit a linear recursion, of constant order when the support of the jump measure is bounded, by means of which they can be calculated—some examples are presented. An application to the modeling of an insurance company’s aggregate capital process is briefly considered. Full article
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