Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Multifractality of jump diffusion processes

Xiaochuan Yang

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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 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.


Nous étudions la régularité locale et la nature multifractale des trajectoires de diffusion à sauts, qui sont solutions d’une classe d’équations stochastiques à sauts. Cet article prolonge et étend substantiellement le travail récent de Barral et al. qui ont construit un processus de Markov de sauts purs avec un spectre multifractal aléatoire. La classe considérée est beaucoup plus large et présente de nouveaux phénomènes multifractals notamment sur les valeurs extrêmes du spectre. Cette classe comprend les processus de type stable au sens de Bass et des EDS non dégénérées guidées par un processus stable.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2042-2074.

Received: 9 October 2015
Revised: 17 August 2017
Accepted: 4 September 2017
First available in Project Euclid: 18 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 28A80: Fractals [See also 37Fxx] 28A78: Hausdorff and packing measures

Jump diffusions Markov processes Stochastic differential equations Hausdorff dimensions Multifractals


Yang, Xiaochuan. Multifractality of jump diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2042--2074. doi:10.1214/17-AIHP864.

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