Refracted Levy processes

Andreas E Kyprianou, R L Loeffen

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64 Citations (SciVal)


Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Levy processes. The latter is a Levy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Levy process is described by the unique strong solution to the stochastic differential equation dU(t) = -delta 1({Ut > b})dt + dX(t), where X = {X-t: t >= 0) is a Levy process with law P and b, delta is an element of R such that the resulting process U may visit the half line (b, infinity) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Levy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.
Original languageEnglish
Pages (from-to)24-44
Number of pages21
JournalAnnales de l'Institut Henri Poincaré: Probabilités et Statistiques
Issue number1
Publication statusPublished - Feb 2010


  • Levy processes
  • Stochastic control
  • fluctuation theory


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