Levy processes: long time behavior and convolution-type form of the Ito representation of the infinitesimal generator

2013/05/18 by Lev Sakhnovich, Sakhnovich, Lev · 1 citation
Mathematics · #45A05 #60G51 #60J45 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Statistics Theory (math.ST) #math.FA #math.PR #math.ST #msc:45A05 #msc:60G51 #msc:60J45 #stat.TH

paper · pdf · doi:10.48550/arxiv.1306.1492

arXiv admin note: text overlap with arXiv:math/0702378, arXiv:1212.3603

arxiv created 2013/05/18 · arxiv updated 2013/06/07

Abstract

In the present paper we show that the Levy-Ito representation of the infinitesimal generator L for Levy processes X_t can be written in a convolution-type form. Using the obtained convolution form we have constructed the quasi-potential operator B. We denote by p(t,Δ) the probability that a sample of the process X_t remains inside the domain Δ for 0≤τ≤t (ruin problem). With the help of the operator B we find a new formula for p(t,Δ). This formula allows us to obtain long time behavior of p(t,Δ).

Citations

• Integral Equations in the Theory of Levy Processes
• Convolution type form of the Ito representation of the infinitesimal generator for Levy processes

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• Levy processes with summable Levy measures, long time behavior

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