Multifractality of jump diffusion processes

2015/02/13 by Xiaochuan Yang, Yang, Xiaochuan
Economics, Econometrics and Finance · Mathematics · #28A78 #28A80 #60H10 #60J25 #60J75 #Complex Systems and Time Series Analysis #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Probability (math.PR) #Stochastic processes and financial applications #math.MG #math.PR #msc:28A78 #msc:28A80 #msc:60H10 #msc:60J25 #msc:60J75

paper · pdf · doi:10.48550/arxiv.1502.03938

33 pages, accepted by Annales de l'Institut Henri Poincaré

openalex publication_date 2015/02/13 · arxiv created 2017/09/05 · arxiv updated 2017/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral \it et al. who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass' stable-like processes and non-degenerate stable-driven SDEs.

Citations

• A Jump Type SDE Approach to Positive Self-Similar Markov Processes
• Fractal Geometry: Mathematical Foundations and Applications.
• Singularities of stable super-Brownian motion
• On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Related

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• The Multifractal Nature of Boltzmann Processes
• Bounds on the support of the multifractal spectrum of stochastic processes
• Intermittency and multifractality: A case study via parabolic stochastic PDEs
• A survey on prescription of multifractal behavior