2008/01/30 by Andreas E. Kyprianou, Kyprianou, Andreas E., Ronnie Loeffen +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60J40 #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.0801.4655
openalex publication_date 2008/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter is a Lévy 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 Lévy process is described by the unique strong solution to the stochastic differential equation \D Ut = - δ1_\Ut gt;b\\D t + \D Xt where X=\Xt :t≥ 0\ is a Lévy process with law ℙ and b, δ∈ ℝ such that the resulting process U may visit the half line (b,∞) 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 Lévy 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.