Space and time inversions of stochastic processes and Kelvin transform

2017/04/04 by Larbi Alili, Loïc Chaumont, Alili, Larbi +5 · 2 citations
Economics, Econometrics and Finance · Mathematics · #31C05 (Primary) #60J45 #60J60 (Secondary) #60J65 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1704.00916

openalex publication_date 2017/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob h-transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.

Cited by

• Chaotic and Predictable Representations for Markov Additive Processes with Levy Modulator
• Non-Archimedean Kelvin Transformation

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