Annals of Applied Probability

Rough path recursions and diffusion approximations

David Kelly

Full-text: Open access


In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and MCMC algorithms. We make no particular probabilistic assumptions on the type of noise appearing in these recursions. Thus, our technique is well suited to recursions where the noise sequence is not a semi-martingale, even though the limiting noise may be. Our main theorem assumes a weak limit theorem on the noise process appearing in the random recursions and lifts it to diffusion approximation for the recursion itself. To achieve this, we approximate the recursion (pathwise) by the solution to a stochastic equation driven by piecewise smooth paths; this can be thought of as a pathwise version of backward error analysis for SDEs. We then identify the limit of this stochastic equation, and hence the original recursion, using tools from rough path theory. We provide several examples of diffusion approximations, both new and old, to illustrate this technique.

Article information

Ann. Appl. Probab., Volume 26, Number 1 (2016), 425-461.

Received: February 2014
Revised: December 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations [See also 34F05] 60H35: Computational methods for stochastic equations [See also 65C30] 39A50: Stochastic difference equations

Stochastic differential equations rough path theory diffusion limits nonsemi-martingale numerical schemes


Kelly, David. Rough path recursions and diffusion approximations. Ann. Appl. Probab. 26 (2016), no. 1, 425--461. doi:10.1214/15-AAP1096.

Export citation


  • [1] Bálint, P. and Melbourne, I. (2008). Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows. J. Stat. Phys. 133 435–447.
  • [2] Breuillard, E., Friz, P. and Huesmann, M. (2009). From random walks to rough paths. Proc. Amer. Math. Soc. 137 3487–3496.
  • [3] Davie, A. M. (2007). Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. AMRX Art. ID abm009, 40.
  • [4] Davydov, Ju. A. (1970). The invariance principle for stationary processes. Teor. Verojatnost. i Primenen. 15 498–509.
  • [5] de Simoi, J. and Liverani, C. (2014). The martingale approach after Varadhan and Dolgopyat. Preprint. Available at arXiv:1402.0090.
  • [6] Debussche, A. and Faou, E. (2012). Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50 1735–1752.
  • [7] Deya, A., Neuenkirch, A. and Tindel, S. (2012). A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 518–550.
  • [8] Dolgopyat, D. (2004). Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 1637–1689 (electronic).
  • [9] Duffie, D. and Protter, P. (1992). From discrete to continuous time finance: Weak convergence of the financial gain process. Math. Finance 2 1–15.
  • [10] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • [11] Friz, P. and Hairer, M. (2014). A Course on Rough Paths. Springer, Berlin.
  • [12] Friz, P. and Riedel, S. (2014). Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 154–194.
  • [13] Friz, P. and Victoir, N. (2008). Euler estimates for rough differential equations. J. Differential Equations 244 388–412.
  • [14] 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.
  • [15] Gottwald, G. A. and Melbourne, I. (2013). Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 20130201, 16.
  • [16] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140.
  • [17] Gubinelli, M. (2010). Ramification of rough paths. J. Differential Equations 248 693–721.
  • [18] Guess, H. A. and Gillespie, J. H. (1977). Diffusion approximations to linear stochastic difference equations with stationary coefficients. J. Appl. Probab. 14 58–74.
  • [19] Holland, M. and Melbourne, I. (2007). Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. (2) 76 345–364.
  • [20] Kelly, D. and Melbourne, I. (2014). Deterministic homogenization for fast slow systems with chaotic noise. Available at arXiv:1409.5748.
  • [21] Kelly, D. and Melbourne, I. (2015). Smooth approximations of stochastic differential equations. Ann. Probab. To appear.
  • [22] Kelly, D. T. B., Law, K. J. H. and Stuart, A. M. (2014). Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. Nonlinearity 27 2579–2604.
  • [23] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
  • [24] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
  • [25] Kushner, H. J. and Huang, H. (1981). On the weak convergence of a sequence of general stochastic difference equations to a diffusion. SIAM J. Appl. Math. 40 528–541.
  • [26] Lamperti, J. (1962). On convergence of stochastic processes. Trans. Amer. Math. Soc. 104 430–435.
  • [27] Lejay, A. and Lyons, T. (2005). On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization. In Current Trends in Potential Theory. Theta Ser. Adv. Math. 4 63–84. Theta, Bucharest.
  • [28] Lejay, A. and Victoir, N. (2006). On $(p,q)$-rough paths. J. Differential Equations 225 103–133.
  • [29] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [30] Majda, A. J., Timofeyev, I. and Vanden Eijnden, E. (2001). A mathematical framework for stochastic climate models. Comm. Pure Appl. Math. 54 891–974.
  • [31] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620–628.
  • [32] Melbourne, I. and Nicol, M. (2009). A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 478–505.
  • [33] Norman, M. F. (1974). Markovian learning processes. SIAM Rev. 16 143–162.
  • [34] Penland, C. and Ewald, B. D. (2008). On modelling physical systems with stochastic models: Diffusion versus Lévy processes. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 2457–2476.
  • [35] Perkowski, N. and Promel, D. J. (2013). Pathwise stochastic integrals for model free finance. Preprint. Available at arXiv:1311.6187.
  • [36] Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110–120.
  • [37] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.
  • [38] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprint of the 1997 edition.
  • [39] Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • [40] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.
  • [41] Watterson, G. A. (1964). The application of diffusion theory to two population genetic models of Moran. J. Appl. Probab. 1 233–246.
  • [42] Young, L.-S. (1998). Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 585–650.
  • [43] Zygalakis, K. C. (2011). On the existence and the applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput. 33 102–130.