The Annals of Probability

Smooth approximation of stochastic differential equations

David Kelly and Ian Melbourne

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Abstract

Consider an Itô process $X$ satisfying the stochastic differential equation $dX=a(X)\,dt+b(X)\,dW$ where $a,b$ are smooth and $W$ is a multidimensional Brownian motion. Suppose that $W_{n}$ has smooth sample paths and that $W_{n}$ converges weakly to $W$. A central question in stochastic analysis is to understand the limiting behavior of solutions $X_{n}$ to the ordinary differential equation $dX_{n}=a(X_{n})\,dt+b(X_{n})\,dW_{n}$.

The classical Wong–Zakai theorem gives sufficient conditions under which $X_{n}$ converges weakly to $X$ provided that the stochastic integral $\int b(X)\,dW$ is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of $\int b(X)\,dW$ depends sensitively on how the smooth approximation $W_{n}$ is chosen.

In applications, a natural class of smooth approximations arise by setting $W_{n}(t)=n^{-1/2}\int_{0}^{nt}v\circ\phi_{s}\,ds$ where $\phi_{t}$ is a flow (generated, e.g., by an ordinary differential equation) and $v$ is a mean zero observable. Under mild conditions on $\phi_{t}$, we give a definitive answer to the interpretation question for the stochastic integral $\int b(X)\,dW$. Our theory applies to Anosov or Axiom A flows $\phi_{t}$, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on $\phi_{t}$.

The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 479-520.

Dates
Received: March 2014
Revised: October 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1454423047

Digital Object Identifier
doi:10.1214/14-AOP979

Mathematical Reviews number (MathSciNet)
MR3456344

Zentralblatt MATH identifier
1372.60082

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Keywords
Interpretation of stochastic integrals Wong–Zakai approximation uniform and nonuniform hyperbolicity rough paths iterated invariance principle

Citation

Kelly, David; Melbourne, Ian. Smooth approximation of stochastic differential equations. Ann. Probab. 44 (2016), no. 1, 479--520. doi:10.1214/14-AOP979. https://projecteuclid.org/euclid.aop/1454423047


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References

  • [1] Alves, J. F. and Azevedo, D. (2013). Statistical properties of diffeomorphisms with weak invariant manifolds. Preprint.
  • [2] Alves, J. F. and Pinheiro, V. (2008). Slow rates of mixing for dynamical systems with hyperbolic structures. J. Stat. Phys. 131 505–534.
  • [3] Anosov, D. V. (1967). Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. 90 1–209.
  • [4] Bálint, P. and Melbourne, I. (2010). Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas. Preprint.
  • [5] Benedicks, M. and Young, L.-S. (2000). Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261 13–56.
  • [6] Bowen, R. (1975). Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics 470. Springer, Berlin.
  • [7] Breuillard, E., Friz, P. and Huesmann, M. (2009). From random walks to rough paths. Proc. Amer. Math. Soc. 137 3487–3496.
  • [8] Brown, B. M. (1971). Martingale central limit theorems. Ann. Math. Statist. 42 59–66.
  • [9] Bunimovich, L. A., Sinaĭ, Y. G. and Chernov, N. I. (1991). Statistical properties of two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk 46 43–92.
  • [10] Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19–42.
  • [11] Denker, M. and Philipp, W. (1984). Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dynam. Systems 4 541–552.
  • [12] Dolgopyat, D. (1998). On decay of correlations in Anosov flows. Ann. of Math. (2) 147 357–390.
  • [13] Dolgopyat, D. (1998). Prevalence of rapid mixing in hyperbolic flows. Ergodic Theory Dynam. Systems 18 1097–1114.
  • [14] Dolgopyat, D. (2004). Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 1637–1689 (electronic).
  • [15] Dolgopyat, D. (2005). Averaging and invariant measures. Mosc. Math. J. 5 537–576, 742.
  • [16] Field, M., Melbourne, I. and Török, A. (2007). Stability of mixing and rapid mixing for hyperbolic flows. Ann. of Math. (2) 166 269–291.
  • [17] 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.
  • [18] Givon, D., Kupferman, R. and Stuart, A. (2004). Extracting macroscopic dynamics: Model problems and algorithms. Nonlinearity 17 R55–R127.
  • [19] Gouëzel, S. (2007). Statistical properties of a skew product with a curve of neutral points. Ergodic Theory Dynam. Systems 27 123–151.
  • [20] Gouëzel, S. (2010). Almost sure invariance principle for dynamical systems by spectral methods. Ann. Probab. 38 1639–1671.
  • [21] Huisinga, W., Schütte, C. and Stuart, A. M. (2003). Extracting macroscopic stochastic dynamics: Model problems. Comm. Pure Appl. Math. 56 234–269.
  • [22] Jakubowski, A., Mémin, J. and Pagès, G. (1989). Convergence en loi des suites d’intégrales stochastiques sur l’espace $\mathbf{D}{}^{1}$ de Skorokhod. Probab. Theory Related Fields 81 111–137.
  • [23] Kelly, D. T. B. (2014). Rough path recursions and diffusion approximations. Preprint.
  • [24] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
  • [25] 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.
  • [26] Lesigne, E. and Volný, D. (2001). Large deviations for martingales. Stochastic Process. Appl. 96 143–159.
  • [27] Liverani, C. (2004). On contact Anosov flows. Ann. of Math. (2) 159 1275–1312.
  • [28] Liverani, C., Saussol, B. and Vaienti, S. (1999). A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 671–685.
  • [29] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [30] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620–628.
  • [31] McShane, E. J. (1972). Stochastic differential equations and models of random processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability Theory 263–294. Univ. California Press, Berkeley, CA.
  • [32] Melbourne, I. (2007). Rapid decay of correlations for nonuniformly hyperbolic flows. Trans. Amer. Math. Soc. 359 2421–2441 (electronic).
  • [33] Melbourne, I. and Nicol, M. (2005). Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 131–146.
  • [34] Melbourne, I. and Nicol, M. (2008). Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 6661–6676.
  • [35] Melbourne, I. and Nicol, M. (2009). A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 478–505.
  • [36] Melbourne, I. and Stuart, A. M. (2011). A note on diffusion limits of chaotic skew-product flows. Nonlinearity 24 1361–1367.
  • [37] Melbourne, I. and Török, A. (2004). Statistical limit theorems for suspension flows. Israel J. Math. 144 191–209.
  • [38] Melbourne, I. and Török, A. (2012). Convergence of moments for Axiom A and nonuniformly hyperbolic flows. Ergodic Theory Dynam. Systems 32 1091–1100.
  • [39] Melbourne, I. and Varandas, P. (2014). A note on statistical properties for nonuniformly hyperbolic systems with subexponential contraction and expansion. Preprint.
  • [40] Melbourne, I. and Zweimüller, R. (2015). Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems. Ann. Inst. Henri Poincaré Probab. Stat. 51 545–556.
  • [41] Móricz, F. (1976). Moment inequalities and the strong laws of large numbers. Z. Wahrsch. Verw. Gebiete 35 299–314.
  • [42] Papanicolaou, G. C. and Kohler, W. (1974). Asymptotic theory of mixing stochastic ordinary differential equations. Comm. Pure Appl. Math. 27 641–668.
  • [43] Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics 53. Springer, New York.
  • [44] Pomeau, Y. and Manneville, P. (1980). Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 189–197.
  • [45] Ratner, M. (1973). The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature. Israel J. Math. 16 181–197.
  • [46] Ruelle, D. (1978). Thermodynamic Formalism. Encyclopedia of Mathematics and Its Applications 5. Addison-Wesley, Reading, MA.
  • [47] Serfling, R. J. (1970). Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41 1227–1234.
  • [48] Sinaĭ, J. G. (1972). Gibbs measures in ergodic theory. Russ. Math. Surv. 27 21–70.
  • [49] Skorohod, A. V. (1956). Limit theorems for stochastic processes. Teor. Veroyatn. Primen. 1 289–319.
  • [50] Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73 747–817.
  • [51] Sussmann, H. J. (1978). On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6 19–41.
  • [52] Sussmann, H. J. (1991). Limits of the Wong–Zakai type with a modified drift term. In Stochastic Analysis 475–493. Academic Press, Boston, MA.
  • [53] Wong, E. and Zakai, M. (1965). On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36 1560–1564.
  • [54] Young, L.-S. (1998). Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 585–650.
  • [55] Young, L.-S. (1999). Recurrence times and rates of mixing. Israel J. Math. 110 153–188.
  • [56] Zweimüller, R. (2007). Mixing limit theorems for ergodic transformations. J. Theoret. Probab. 20 1059–1071.