Annals of Probability

General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula

Peter K. Friz and Atul Shekhar

Full-text: Open access


We consider rough paths with jumps. In particular, the analogue of Lyons’ extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against càdlàg processes. A class of Lévy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and a certain integrability property of the Carnot–Caratheodory norm with respect to the Lévy measure on the group, using Hunt’s framework of Lie group valued Lévy processes. Examples of Lévy rough paths include a standard multi-dimensional Lévy process enhanced with a stochastic area as constructed by D. Williams, the pure area Poisson process and Brownian motion in a magnetic field. An explicit formula for the expected signature is given.

Article information

Ann. Probab., Volume 45, Number 4 (2017), 2707-2765.

Received: January 2015
Revised: June 2016
First available in Project Euclid: 11 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H99: None of the above, but in this section

Young integration rough paths Lévy processes general theory of processes


Friz, Peter K.; Shekhar, Atul. General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula. Ann. Probab. 45 (2017), no. 4, 2707--2765. doi:10.1214/16-AOP1123.

Export citation


  • [1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 93. Cambridge Univ. Press, Cambridge.
  • [2] Applebaum, D. and Kunita, H. (1993). Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33 1103–1123.
  • [3] Baudoin, F. (2004). An Introduction to the Geometry of Stochastic Flows. Imperial College Press, London.
  • [4] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [5] Boedihardjo, H., Geng, X., Lyons, T. and Yang, D. (2016). The signature of a rough path: Uniqueness. Adv. Math. 293 720–737.
  • [6] Chevyrev, I. (2015). Random walks and Lévy processes as rough paths. Probab. Theory Relat. Fields. To appear. Available at arXiv:1510.09066.
  • [7] Chevyrev, I. and Lyons, T. (2016). Characteristic functions of measures on geometric rough paths. Ann. Probab. 44 4049–4082.
  • [8] Dudley, R. M. and Norvaiša, R. (1998). An Introduction to $P$-variation and Young Integrals: With Emphasis on Sample Functions of Stochastic Processes. Lecture Notes. MaPhySto, Dept. Mathematical Sciences, Univ. Aarhus.
  • [9] Fawcett, T. A. (2002). Non-commutative harmonic analysis. Ph.D. thesis, Mathematical Institute, Univ. Oxford.
  • [10] Föllmer, H. (1981). Calcul d’Itô sans probabilités. In Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French). Lecture Notes in Math. 850 143–150. Springer, Berlin.
  • [11] Friz, P., Gassiat, P. and Lyons, T. (2015). Physical Brownian motion in a magnetic field as a rough path. Trans. Amer. Math. Soc. 367 7939–7955.
  • [12] Friz, P. and Victoir, N. (2006). A variation embedding theorem and applications. J. Funct. Anal. 239 631–637.
  • [13] Friz, P. and Victoir, N. (2008). The Burkholder–Davis–Gundy inequality for enhanced martingales. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 421–438. Springer, Berlin.
  • [14] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths. With an introduction to regularity structures. Springer, Cham.
  • [15] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [16] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140.
  • [17] Hambly, B. and Lyons, T. (2010). Uniqueness for the signature of a path of bounded variation and the reduced path group. Ann. of Math. (2) 171 109–167.
  • [18] Hunt, G. A. (1956). Semi-groups of measures on Lie groups. Trans. Amer. Math. Soc. 81 264–293.
  • [19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [20] Jain, N. C. and Monrad, D. (1983). Gaussian measures in $B_{p}$. Ann. Probab. 11 46–57.
  • [21] Karandikar, R. L. (1995). On pathwise stochastic integration. Stochastic Process. Appl. 57 11–18.
  • [22] Kurtz, T. G., Pardoux, É. and Protter, P. (1995). Stratonovich stochastic differential equations driven by general semimartingales. Ann. Inst. Henri Poincaré Probab. Stat. 31 351–377.
  • [23] Ledoux, M., Lyons, T. and Qian, Z. (2002). Lévy area of Wiener processes in Banach spaces. Ann. Probab. 30 546–578.
  • [24] Lépingle, D. (1976). La variation d’ordre $p$ des semi-martingales. Z. Wahrsch. Verw. Gebiete 36 295–316.
  • [25] Le Jan, Y. and Qian, Z. (2013). Stratonovich’s signatures of Brownian motion determine Brownian sample paths. Probab. Theory Related Fields 157 209–223.
  • [26] Liao, M. (2004). Lévy Processes in Lie Groups. Cambridge Tracts in Mathematics 162. Cambridge Univ. Press, Cambridge.
  • [27] Lyons, T. and Ni, H. (2015). Expected signature of Brownian motion up to the first exit time from a bounded domain. Ann. Probab. 43 2729–2762.
  • [28] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Univ. Press, Oxford. Oxford Science Publications.
  • [29] Lyons, T. and Victoir, N. (2004). Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 169–198. Stochastic analysis with applications to mathematical finance.
  • [30] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [31] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [32] Manstavičius, M. (2004). $p$-variation of strong Markov processes. Ann. Probab. 32 2053–2066.
  • [33] Marcus, S. I. (1978). Modeling and analysis of stochastic differential equations driven by point processes. IEEE Trans. Inform. Theory 24 164–172.
  • [34] Marcus, S. I. (1980/81). Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics 4 223–245.
  • [35] Mikosch, T. and Norvaiša, R. (2000). Stochastic integral equations without probability. Bernoulli 6 401–434.
  • [36] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Stat. 43 1293–1311.
  • [37] Ni, H. (2013). The Expected Signature of a Stochastic Process. Ph.D. thesis, Mathematical Institute, Univ. Oxford.
  • [38] Perkowski, N. and Prömel, D. J. (2016). Pathwise stochastic integrals for model free finance. Bernoulli 22 2486–2520.
  • [39] Prömel, D. J. and Trabs, M. (2016). Rough differential equations driven by signals in Besov spaces. J. Differential Equations 260 5202–5249.
  • [40] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed., version 2.1, corrected third printing. Stochastic Modelling and Applied Probability 21. Springer, Berlin.
  • [41] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Itô calculus, Reprint of the second (1994) edition.
  • [42] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. Translated from the 1990 Japanese original, revised by the author.
  • [43] Simon, T. (2003). Small deviations in $p$-variation for multidimensional Lévy processes. J. Math. Kyoto Univ. 43 523–565.
  • [44] Soner, H. M., Touzi, N. and Zhang, J. (2011). Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 paper no. 67, 1844–1879.
  • [45] Stroock, D. W. (1975). Diffusion processes associated with Lévy generators. Z. Wahrsch. Verw. Gebiete 32 209–244.
  • [46] Williams, D. R. E. (2001). Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoam. 17 295–329.
  • [47] Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 251–282.