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: closed (30 September 2019).

Special Issue Editor

Prof. Dr. Florin Avram
Website
Guest Editor
Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France
Interests: stochastic processes; risk; mathematical finance; inventory; queueing and population dynamics
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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

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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 (12 papers)

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Research

Open AccessArticle
On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes
Risks 2019, 7(4), 121; https://doi.org/10.3390/risks7040121 - 11 Dec 2019
Abstract
This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a [...] Read more.
This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called “shifted” Padé approximation. Full article
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Open AccessArticle
The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps
Risks 2019, 7(4), 120; https://doi.org/10.3390/risks7040120 - 10 Dec 2019
Abstract
In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Løkka and Zervos (2008); Lindensjö and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and [...] Read more.
In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Løkka and Zervos (2008); Lindensjö and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and Yang (2016), for diffusion processes, we will show that the so-called Løkka–Zervos alternative also holds true in the case of a Cramér–Lundberg risk process with exponential claims. More specifically, we show that: if the cost of capital injections is low, then according to a double-barrier strategy, it is optimal to pay dividends and inject capital, meaning ruin never occurs; and if the cost of capital injections is high, then according to a single-barrier strategy, it is optimal to pay dividends and never inject capital, meaning ruin occurs at the first passage below zero. Full article
Open AccessArticle
A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems
Risks 2019, 7(4), 117; https://doi.org/10.3390/risks7040117 - 19 Nov 2019
Abstract
The Segerdahl-Tichy Process, characterized by exponential claims and state dependent drift, has drawn a considerable amount of interest, due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest [...] Read more.
The Segerdahl-Tichy Process, characterized by exponential claims and state dependent drift, has drawn a considerable amount of interest, due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest non-Lévy, non-diffusion example of a spectrally negative Markov risk model. Note that for both spectrally negative Lévy and diffusion processes, first passage theories which are based on identifying two “basic” monotone harmonic functions/martingales have been developed. This means that for these processes many control problems involving dividends, capital injections, etc., may be solved explicitly once the two basic functions have been obtained. Furthermore, extensions to general spectrally negative Markov processes are possible; unfortunately, methods for computing the basic functions are still lacking outside the Lévy and diffusion classes. This divergence between theoretical and numerical is strikingly illustrated by the Segerdahl process, for which there exist today six theoretical approaches, but for which almost nothing has been computed, with the exception of the ruin probability. Below, we review four of these methods, with the purpose of drawing attention to connections between them, to underline open problems, and to stimulate further work. Full article
Open AccessArticle
Three Essays on Stopping
Risks 2019, 7(4), 105; https://doi.org/10.3390/risks7040105 - 18 Oct 2019
Abstract
First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if [...] Read more.
First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017). Full article
Open AccessFeature PaperArticle
Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims
Risks 2019, 7(4), 104; https://doi.org/10.3390/risks7040104 - 14 Oct 2019
Cited by 1
Abstract
We consider the Sparre Andersen risk process with interclaim times that belong to the class of distributions with rational Laplace transform. We construct error bounds for the ruin probability based on the Pollaczek–Khintchine formula, and develop an efficient algorithm to approximate the ruin [...] Read more.
We consider the Sparre Andersen risk process with interclaim times that belong to the class of distributions with rational Laplace transform. We construct error bounds for the ruin probability based on the Pollaczek–Khintchine formula, and develop an efficient algorithm to approximate the ruin probability for completely monotone claim size distributions. Our algorithm improves earlier results and can be tailored towards achieving a predetermined accuracy of the approximation. Full article
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Open AccessArticle
On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes
Risks 2019, 7(3), 87; https://doi.org/10.3390/risks7030087 - 05 Aug 2019
Abstract
We obtain closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown or drawup process hits a constant level before an independent exponential [...] Read more.
We obtain closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown or drawup process hits a constant level before an independent exponential random time. It is assumed that the coefficients of the diffusion-type process are regular functions of the current values of its running maximum and minimum. The proof is based on the solution to the equivalent inhomogeneous ordinary differential boundary-value problem and the application of the normal-reflection conditions for the value function at the edges of the state space of the resulting three-dimensional Markov process. The result is related to the computation of probability characteristics of the take-profit and stop-loss values of a market trader during a given time period. Full article
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Open AccessArticle
Potential Densities for Taxed Spectrally Negative Lévy Risk Processes
Risks 2019, 7(3), 85; https://doi.org/10.3390/risks7030085 - 02 Aug 2019
Abstract
This paper revisits the spectrally negative Lévy risk process embedded with the general tax structure introduced in Kyprianou and Zhou (2009). A joint Laplace transform is found concerning the first down-crossing time below level 0. The potential density is also obtained for the [...] Read more.
This paper revisits the spectrally negative Lévy risk process embedded with the general tax structure introduced in Kyprianou and Zhou (2009). A joint Laplace transform is found concerning the first down-crossing time below level 0. The potential density is also obtained for the taxed Lévy risk process killed upon leaving [ 0 , b ] . The results are expressed using scale functions. Full article
Open AccessArticle
Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model
Risks 2019, 7(3), 83; https://doi.org/10.3390/risks7030083 - 01 Aug 2019
Abstract
We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function P(u) for the component-wise ruin (that is both business lines are ruined [...] Read more.
We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function P ( u ) for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where u is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of ln P ( u ) / u as u tends to infinity, which depends essentially on the correlation ρ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem. Full article
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Open AccessArticle
De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes
Risks 2019, 7(3), 73; https://doi.org/10.3390/risks7030073 - 03 Jul 2019
Cited by 1
Abstract
We consider de Finetti’s stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show [...] Read more.
We consider de Finetti’s stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show that, under mild assumptions on the Lévy measure, an optimal strategy is formed by a barrier strategy and that this optimal barrier level is always less than the optimal barrier level when classical ruin is implemented. In addition, we give necessary and sufficient conditions for the barrier strategy at level zero to be optimal. Full article
Open AccessArticle
The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps
Risks 2019, 7(1), 18; https://doi.org/10.3390/risks7010018 - 19 Feb 2019
Cited by 2
Abstract
As is well-known, the benefit of restricting Lévy processes without positive jumps is the “W,Z scale functions paradigm”, by which the knowledge of the scale functions W,Z extends immediately to other risk control problems. The same is true [...] Read more.
As is well-known, the benefit of restricting Lévy processes without positive jumps is the “ W , Z scale functions paradigm”, by which the knowledge of the scale functions W , Z extends immediately to other risk control problems. The same is true largely for strong Markov processes X t , with the notable distinctions that (a) it is more convenient to use as “basis” differential exit functions ν , δ , and that (b) it is not yet known how to compute ν , δ or W , Z beyond the Lévy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Lévy cases are very similar (except for the level of work involved in computing the basic functions ν , δ ). We illustrate the potential of the unified framework by introducing a new objective (33) for the optimization of dividends, inspired by the de Finetti problem of maximizing expected discounted cumulative dividends until ruin, where we replace ruin with an optimally chosen Azema-Yor/generalized draw-down/regret/trailing stopping time. This is defined as a hitting time of the “draw-down” process Y t = sup 0 s t X s X t obtained by reflecting X t at its maximum. This new variational problem has been solved in a parallel paper. Full article
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Open AccessArticle
Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint
Risks 2019, 7(1), 13; https://doi.org/10.3390/risks7010013 - 31 Jan 2019
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
Risks 2018, 6(3), 102; https://doi.org/10.3390/risks6030102 - 18 Sep 2018
Cited by 1
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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