Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Multifractality of jump diffusion processes

Xiaochuan Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849792

Digital Object Identifier
doi:10.1214/17-AIHP864

Mathematical Reviews number (MathSciNet)
MR3865666

Zentralblatt MATH identifier
06996558

Subjects
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

Keywords
Jump diffusions Markov processes Stochastic differential equations Hausdorff dimensions Multifractals

Citation

Yang, Xiaochuan. Multifractality of jump diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2042--2074. doi:10.1214/17-AIHP864. https://projecteuclid.org/euclid.aihp/1539849792


Export citation

References

  • [1] P. Abry, H. Jaffard and H. Wendt. Irregularities and Scaling in Signal and Image Processing: Multifractal Analysis. F. Michael. (Ed.) Benoit Mandelbrot: A Life in Many Dimensions, 31–116. World Scientific, Singapore, 2015.
  • [2] K. I. Amin. Jump diffusion option valuation in discrete time. J. Finance 48 (5) (1993) 1833–1863.
  • [3] D. Applebaum. Lévy Processes and Stochastic Calculus, 2nd edition. Cambridge Studies in Advanced Mathematics 116. Cambridge University Press, Cambridge, 2009.
  • [4] A. Ayache and M. S. Taqqu. Multifractional processes with random exponent. Publ. Mat. 49 (2) (2005) 459–486.
  • [5] P. Balança. Fine regularity of Lévy processes and linear (multi)fractional stable motion. Electron. J. Probab. 19, 101, 37 (2014).
  • [6] P. Balança. Some sample path properties of multifractional Brownian motion. Stochastic Process. Appl. 125 (10) (2015) 3823–3850.
  • [7] P. Balança and L. Mytnik. Singularities of stable super-brownian motion, (2016). Available at arXiv:1608.00792.
  • [8] M. Barczy, Z. Li and G. Pap. Yamada–Watanabe results for stochastic differential equations with jumps. Int. J. Stoch. Anal. Art. ID 460472, 23 (2015).
  • [9] J. Barral, A. Durand, S. Jaffard and S. Seuret. Local multifractal analysis. In Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II 31–64. Fractals in Applied Mathematics 601. Amer. Math. Soc., Providence, RI, 2013.
  • [10] J. Barral, N. Fournier, S. Jaffard and S. Seuret. A pure jump Markov process with a random singularity spectrum. Ann. Probab. 38 (5) (2010) 1924–1946.
  • [11] J. Barral and S. Seuret. The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 214 (1) (2007) 437–468.
  • [12] J. Barral and S. Seuret. A localized Jarník–Besicovitch theorem. Adv. Math. 226 (4) (2011) 3191–3215.
  • [13] R. F. Bass. Uniqueness in law for pure jump Markov processes. Probab. Theory Related Fields 79 (2) (1988) 271–287.
  • [14] R. F. Bass. Stochastic differential equations driven by symmetric stable processes. In Séminaire de Probabilités, XXXVI 302–313. Lecture Notes in Math. 1801. Springer, Berlin, 2003.
  • [15] J. Bertoin. On nowhere differentiability for Lévy processes. Stoch. Stoch. Rep. 50 (3–4) (1994) 205–210.
  • [16] C. T. Chudley and R. J. Elliott. Neutron scattering from a liquid on a jump diffusion model. Proc. Phys. Soc. 77 (2) (1961) 353.
  • [17] E. Çinlar and J. Jacod. Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 159–242. Evanston, Ill., 1981. Progr. Prob. Statist. 1. Birkhäuser, Boston, MA, 1981.
  • [18] L. Döring and M. Barczy. A jump type SDE approach to positive self-similar Markov processes. Electron. J. Probab. 17, 94, 39 (2012).
  • [19] A. Durand. Random wavelet series based on a tree-indexed Markov chain. Comm. Math. Phys. 283 (2) (2008) 451–477.
  • [20] A. Durand. Singularity sets of Lévy processes. Probab. Theory Related Fields 143 (3–4) (2009) 517–544.
  • [21] A. Durand and S. Jaffard. Multifractal analysis of Lévy fields. Probab. Theory Related Fields 153 (1–2) (2012) 45–96.
  • [22] A. Dvoretzky. On the oscillation of the Brownian motion process. Israel J. Math. 1 (1963) 212–214.
  • [23] K. Falconer. Fractal Geometry, 2nd edition. Wiley, Hoboken, NJ, 2003. Mathematical foundations and applications.
  • [24] K. J. Falconer. The local structure of random processes. J. Lond. Math. Soc. (2) 67 (3) (2003) 657–672.
  • [25] N. Fournier. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 (1) (2013) 138–159.
  • [26] Z. Fu and Z. Li. Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120 (3) (2010) 306–330.
  • [27] E. Herbin. From $N$ parameter fractional Brownian motions to $N$ parameter multifractional Brownian motions. Rocky Mountain J. Math. 36 (4) (2006) 1249–1284.
  • [28] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland Mathematical Library 24. North-Holland, Amsterdam, 1989.
  • [29] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin, 2003.
  • [30] S. Jaffard. Old friends revisited: The multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 (1) (1997) 1–22.
  • [31] S. Jaffard. The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 (2) (1999) 207–227.
  • [32] S. Jaffard. Wavelet techniques in multifractal analysis. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 91–151. Proc. Sympos. Pure Math. 72. Amer. Math. Soc., Providence, RI, 2004.
  • [33] D. Khoshnevisan and Z. Shi. Fast sets and points for fractional Brownian motion. In Séminaire de Probabilités, XXXIV 393–416. Lecture Notes in Math. 1729. Springer, Berlin, 2000.
  • [34] T. G. Kurtz. Equivalence of stochastic equations and martingale problems. In Stochastic Analysis 2010 113–130. Springer, Heidelberg, 2011.
  • [35] Z. Li and L. Mytnik. Strong solutions for stochastic differential equations with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 47 (4) (2011) 1055–1067.
  • [36] L. Mytnik and V. Wachtel. Multifractal analysis of superprocesses with stable branching in dimension one. Ann. Probab. 43 (5) (2015) 2763–2809.
  • [37] S. Orey and S. J. Taylor. How often on a Brownian path does the law of iterated logarithm fail? Proc. London Math. Soc. 3 (28) (1974) 174–192.
  • [38] E. Perkins. On the Hausdorff dimension of the Brownian slow points. Z. Wahrsch. Verw. Gebiete 64 (3) (1983) 369–399.
  • [39] E. A. Perkins and S. J. Taylor. The multifractal structure of super-Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 34 (1) (1998) 97–138.
  • [40] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999.
  • [41] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, Cambridge, 2013. Translated from the 1990 Japanese original, Revised edition of the 1999 English translation.
  • [42] L. A. Shepp. Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 23 (1972) 163–170.
  • [43] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005. Mathematical and analytical techniques with applications to engineering.
  • [44] X. Yang. Dimension study of the regularity of jump diffusion processes. Ph.D. Thesis (2016).
  • [45] X. Yang. Hausdorff dimension of the range and the graph of stable-like processes. J. Theor. Probab. (2017).