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.
Highlights
Introduction and main resultsThis article concerns the pointwise regularity of the sample paths of Markov processes
We investigate here the multifractal structure of a quite general class of one-dimensional Markov processes defined by stochastic differential equations with jumps, called jump diffusions: t t
This work is in the line of a large and recent literature investigating the multifractal nature of “Lévy-like” processes [5,10,11,20,21,31], and in particular, aims to generalize the recent work [31] of Jaffard on Lévy processes to a larger class of Markov processes
Summary
This article concerns the pointwise regularity of the sample paths of Markov processes. As an attempt beyond the scope of Lévy processes, Barral–Fournier–Jaffard–Seuret [10] constructed a specific example of pure jump increasing Markov process with a random multifractal spectrum. The specific structure of this process (and in particular the monotonicity of sample paths) simplifies greatly the study of its regularity, and as a consequence, the present work requires more technicality and additional tools. A slicing argument is developed to give some technical increments estimates This new argument does not rely on the monotonicity of the sample paths, is applicable to more general SDEs, see Section 3. There is a novel discussion on the extreme value of the spectrum, the latter presenting a behavior more complex than the one observed in [10], see Section 6
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
