The Annals of Probability

Integrability and tail estimates for Gaussian rough differential equations

Thomas Cass, Christian Litterer, and Terry Lyons

Full-text: Open access


We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter $H>1/4$. We remark on the relevance of such estimates to a number of significant open problems.

Article information

Ann. Probab., Volume 41, Number 4 (2013), 3026-3050.

First available in Project Euclid: 3 July 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60G15: Gaussian processes

Rough path analysis Gaussian processes


Cass, Thomas; Litterer, Christian; Lyons, Terry. Integrability and tail estimates for Gaussian rough differential equations. Ann. Probab. 41 (2013), no. 4, 3026--3050. doi:10.1214/12-AOP821.

Export citation


  • [1] Baudoin, F. and Hairer, M. (2007). A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 373–395.
  • [2] Cass, T. and Friz, P. (2010). Densities for rough differential equations under Hörmander’s condition. Ann. of Math. (2) 171 2115–2141.
  • [3] Cass, T., Friz, P. and Victoir, N. (2009). Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 3359–3371.
  • [4] Cass, T. and Lyons, T. (2010). Evolving communities and individual preferences. Unpublished manuscript.
  • [5] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108–140.
  • [6] Friz, P. and Oberhauser, H. (2009). Rough path limits of the Wong–Zakai type with a modified drift term. J. Funct. Anal. 256 3236–3256.
  • [7] Friz, P. and Oberhauser, H. (2010). A generalized Fernique theorem and applications. Proc. Amer. Math. Soc. 138 3679–3688.
  • [8] Friz, P. and Riedel, S. (2012). Integrability of linear rough differential equations. Available at arXiv:1104.0577v3.
  • [9] Friz, P. and Victoir, N. (2006). A variation embedding theorem and applications. J. Funct. Anal. 239 631–637.
  • [10] Friz, P. and Victoir, N. (2010). Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 369–413.
  • [11] Friz, P. and Victoir, N. (2011). A note on higher dimensional $p$-variation. Electron. J. Probab. 16 1880–1899.
  • [12] 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.
  • [13] Ganesh, A., O’Connell, N. and Wischik, D. (2004). Big Queues. Lecture Notes in Math. 1838. Springer, Berlin.
  • [14] Guasoni, P. (2006). No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 569–582.
  • [15] Gubinelli, M., Lejay, A. and Antipolis, S. (2009). Global existence for rough differential equations under linear growth condition. Available at arXiv:0905.2399v1.
  • [16] Hairer, M. and Mattingly, J. C. (2006). Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 993–1032.
  • [17] Hairer, M. and Pillai, N. (2011). Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Available at arXiv:1104.5218v1.
  • [18] Hairer, M. and Pillai, N. S. (2011). Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 47 601–628.
  • [19] Hambly, B. M. and Lyons, T. J. (1998). Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab. 26 132–148.
  • [20] Hu, Y. and Tindel, S. (2011). Smooth density for some nilpotent rough differential equations. Available at arXiv:1104.1972.
  • [21] Inahama, Y. (2012). A moment estimate of the derivative process in rough path theory. Proc. Amer. Math. Soc. 140 2183–2191.
  • [22] Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 165–294. Springer, Berlin.
  • [23] Ledoux, M., Qian, Z. and Zhang, T. (2002). Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102 265–283.
  • [24] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Univ. Press, Oxford.
  • [25] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [26] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [27] Norris, J. (1986). Simplified Malliavin calculus. In Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math. 1204 101–130. Springer, Berlin.
  • [28] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.