Markov chain approximations to scale functions of Lévy processes

Abstract

(2015) Markov chain approximations to scale functions of Lévy processes. Stochastic Processes and their Applications, 125 (10). pp. 3932-3957. Permanent WRAP url:http://wrap.warwick.ac.uk/75698 Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-forprofit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.

On future drawdowns of Lévy processes

“…This agrees with the forms of L D (r, s) and L U (r, s) obtained in [9]. (iii) In the literature numerical methods have been developed for the evaluation of scale functions, based on Markov chain approximation (see [22]) or Laplace inversion (see [18,30]), which may be used for numerical evaluation of the expressions given in Proposition 5.1. (iv) From the proofs of the propositions above it is clear that we can identify the bivariate Laplace transform of U * t,s , U * t,s , D * t,s and D * t,s with respect of t and s as long as the laws of X eq , X eq and resolvents of reflected process R U a , R D a are known.…”

First passage problems for upwards skip-free random walks via the scale functions paradigm

“…lim y→∞ W (q) (y)e −Φ(q)(y+h) = 1/ψ (Φ(q)+) lim y→∞ W (q) (y)e −Φ(q)(y) = 1/ψ (Φ(q)+) Remark 49. Every spectrally negative Lévy process may be seen as a (weak) limit of a net Y h of upwards skip-free Lévy chains, as h ↓ 0 [MVJ15]. This means that a great many relations in the spectrally negative Lévy setting may be got (at least naively) by simply passing to the limit h ↓ 0 (formally, one must of course pay attention to whether or not the relevant functional is continuous with respect to such a weak limit).…”

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

“…This can be achieved by taking λ({h}) sufficiently large: we will then be witnessing the arrival of premia with very high-intensity, which by the law of large numbers on a large enough time scale will look essentially like premium drift (but slightly stochastic), interdispersed with the arrivals of claims. This is basically an approximation of the Cramér-Lundberg model in the spirit of Mijatović et al (2015), which however (because we are not ultimately effecting the limits h ↓ 0, λ({h}) → ∞) retains some stochasticity in the premia. Keeping this in mind, it would be interesting to see how the upwards skip-free model behaves when fitted against real data, but this investigation lies beyond the intended scope of the present text.…”

“…It was shown in Vidmar (2015) that precisely two types of Lévy processes exhibit the property of non-random overshoots: those with no positive jumps a.s., and compound Poisson processes, whose jump chain is (for some h > 0) a random walk on Z h := {hk: k ∈ Z}, skip-free to the right. The latter class was then referred to as "upwards skip-free Lévy chains".…”

“…Besides, our chains feature as a natural continuous-time approximation of the more subtle spectrally negative Lévy family, that, because of its overall tractability, has been used extensively in applied probability (in particular to model the risk process of an insurance company; see the papers ; Chiu and Yin (2005); Yang and Zhang (2001) among others). This approximation point of view is developed in Mijatović et al (2014Mijatović et al ( , 2015. Finally, focusing on the insurance context, the chains may be used directly to model the aggregate capital process of an insurance company, in what is a continuous-time embedding of the discrete-time compound binomial risk model (for which see ; Bao and Liu (2012); Wat et al (2018); Xiao and Guo (2007) and the references therein).…”