Electronic Communications in Probability

Recovering the pathwise Itô solution from averaged Stratonovich solutions

Terry Lyons and Danyu Yang

Full-text: Open access


We recover the pathwise Itô solution (the solution to a rough differential equation driven by the Itô signature) by concatenating averaged Stratonovich solutions on small intervals and by letting the mesh of the partition in the approximations tend to zero. More specifically, on a fixed small interval, we consider two Stratonovich solutions: one is driven by the original process and the other is driven by the original process plus a selected independent noise. Then by taking the expectation with respect to the selected noise, we can recover the increment of the bracket process and so recover the leading order approximation of the Itô solution up to a small error. By concatenating averaged increments and by letting the mesh tend to zero, the error tends to zero and we recover the Itô solution.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 7, 18 pp.

Received: 14 September 2014
Accepted: 10 August 2015
First available in Project Euclid: 5 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G17: Sample path properties

rough paths theory pathwise Itô solution Stratonovich solution

Creative Commons Attribution 4.0 International License.


Lyons, Terry; Yang, Danyu. Recovering the pathwise Itô solution from averaged Stratonovich solutions. Electron. Commun. Probab. 21 (2016), paper no. 7, 18 pp. doi:10.1214/16-ECP3795. https://projecteuclid.org/euclid.ecp/1454682823

Export citation


  • [1] Bichteler, K., Stochastic integration and $L^{p}$-theory of stochastic integration, Ann. Prob., 9, 48-89, (1981).
  • [2] Coutin, L., Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motion. Probab. Theory Related Fields, 122(1),108-140, (2002).
  • [3] Davie, A. M., Differential equations driven by rough paths: an approach via discrete approximation. Applied Mathematics Research eXpress, abm009, (2008).
  • [4] Föllmer, H., Calcul d’Itô sans probabilités, Seminaire de probabilites (Strasbourg), 15, 143-150, (1981).
  • [5] Friz, P., Oberhauser, H., Rough path limits of the Wong–Zakai type with a modified drift term. J. Funct. Anal., 256(10), 3236-3256, (2009).
  • [6] Friz, P., Victoir, N., Differential equations driven by Gaussian signals. Ann. Inst. H. Poincaré Probab. Statist., 46(2), 369-413, (2010).
  • [7] Friz, P., Victoir, N., Multidimensional Stochastic Processes as Rough Paths, Theory and Applications, Cambridge Univ. Press, (2010).
  • [8] Gubinelli, M., Controlling rough paths, J. Functional Analysis, 216(1), 86-140, (2004).
  • [9] Hairer, M., Kelly, D., Geometric versus non-geometric rough paths, Ann. Inst. H. Poincaré Probab. Statist., 51(1), 207-251, (2015).
  • [10] Hairer, M., Weber, H., Rough Burgers-like equations with multiplicative noise, Probab. Theory Related Fields, 155(1), 71-126, (2013).
  • [11] Itô, K., Stochastic integral, Proc. Imp. Acad. Tokyo, 20, 519-524, (1944).
  • [12] Itô, K., On stochastic differential equations. Proc. Imp. Acad. Tokyo, 22, 32-35, (1946).
  • [13] Karandikar, R. L., Pathwise solution of stochastic differential equations, Sankhya A, 43, 121-132, (1981).
  • [14] Lejay, A., Victoir, N., On $\left ( p,q\right ) $-rough paths, Journal of Differential Equations, 225, 103-133, (2006).
  • [15] Lyons, T. J., Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14(2), 215-310, (1998).
  • [16] Lyons, T. J., Caruana, M., Lévy, T., Picard, J., Differential equation driven by rough paths. Springer, (2007).
  • [17] Lyons, T. J., Qian, Z. M., Calculus for multiplicative functionals, Itô’s lemma and differential equations, Itô’s Stochastic Calculus and Probability Theory, Springer, Tokyo, 233-250, (1996).
  • [18] Lyons, T. J., Qian., Z., System control and rough paths, OxfordUniv. Press, (2002).
  • [19] Lyons, T. J., Stoica, L., The limits of stochastic integrals of differential forms. Ann. Probab., 27(1), 1-49, (1999).
  • [20] Lyons, T. J., Yang, D., The partial sum process of orthogonal expansion as geometric rough process with Fourier series as an example—an improvement of Menshov-Rademacher theorem, J. Functional Analysis, 265 (12), 3067-3103, (2013).
  • [21] Russo, F., Vallois, P., Intégrales progressive, rétrograde et symétrique de processus nonadaptés. C. R. Acad. Sci. Paris Ser. I Math., 312(8), 615-618, (1991).
  • [22] Strook, D. W., Varadhan, S. R. S., On the support of diffusion processes with applications to the strong maximum principle, Proc. Sixth Berkeley Symp. on math. Statist. and Prob., Vol 3, Univ. of Calif. Press, 333-359, (1972).
  • [23] Wong, E., Zakai, M., On the relation between ordinary and stochastic differential equations, Int. J. Engineering Sci., 3(2), 213-229, (1965).
  • [24] Young, L. C., An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67, 251-282, (1936).