The Annals of Applied Probability

Rough path recursions and diffusion approximations

David Kelly

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003244

Digital Object Identifier
doi:10.1214/15-AAP1096

Mathematical Reviews number (MathSciNet)
MR3449323

Zentralblatt MATH identifier
1335.60093

Subjects
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

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

Citation

Kelly, David. Rough path recursions and diffusion approximations. Ann. Appl. Probab. 26 (2016), no. 1, 425--461. doi:10.1214/15-AAP1096. https://projecteuclid.org/euclid.aoap/1452003244


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References

  • [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.
    Mathematical Reviews (MathSciNet): MR2448631
    Zentralblatt MATH: 1161.82016
    Digital Object Identifier: doi:10.1007/s10955-008-9623-y
  • [2] Breuillard, E., Friz, P. and Huesmann, M. (2009). From random walks to rough paths. Proc. Amer. Math. Soc. 137 3487–3496.
    Mathematical Reviews (MathSciNet): MR2515418
    Zentralblatt MATH: 1179.60017
    Digital Object Identifier: doi:10.1090/S0002-9939-09-09930-4
  • [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.
    Mathematical Reviews (MathSciNet): MR2387018
    Zentralblatt MATH: 1163.34005
  • [4] Davydov, Ju. A. (1970). The invariance principle for stationary processes. Teor. Verojatnost. i Primenen. 15 498–509.
    Mathematical Reviews (MathSciNet): MR283872
  • [5] de Simoi, J. and Liverani, C. (2014). The martingale approach after Varadhan and Dolgopyat. Preprint. Available at arXiv:1402.0090.
    arXiv: 1402.0090
  • [6] Debussche, A. and Faou, E. (2012). Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50 1735–1752.
    Mathematical Reviews (MathSciNet): MR2970763
    Zentralblatt MATH: 1256.65002
    Digital Object Identifier: doi:10.1137/110831544
  • [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.
    Mathematical Reviews (MathSciNet): MR2954265
    Digital Object Identifier: doi:10.1214/10-AIHP392
    Project Euclid: euclid.aihp/1334148209
  • [8] Dolgopyat, D. (2004). Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 1637–1689 (electronic).
    Mathematical Reviews (MathSciNet): MR2034323
    Zentralblatt MATH: 1031.37031
    Digital Object Identifier: doi:10.1090/S0002-9947-03-03335-X
  • [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.
    Mathematical Reviews (MathSciNet): MR1964455
  • [11] Friz, P. and Hairer, M. (2014). A Course on Rough Paths. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR3289027
    Zentralblatt MATH: 06323240
  • [12] Friz, P. and Riedel, S. (2014). Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 154–194.
    Mathematical Reviews (MathSciNet): MR3161527
    Zentralblatt MATH: 1295.60045
    Digital Object Identifier: doi:10.1214/12-AIHP507
    Project Euclid: euclid.aihp/1388545270
  • [13] Friz, P. and Victoir, N. (2008). Euler estimates for rough differential equations. J. Differential Equations 244 388–412.
    Mathematical Reviews (MathSciNet): MR2376201
    Zentralblatt MATH: 1140.60037
    Digital Object Identifier: doi:10.1016/j.jde.2007.10.008
  • [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.
    Mathematical Reviews (MathSciNet): MR2604669
    Zentralblatt MATH: 1193.60053
  • [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.
    Mathematical Reviews (MathSciNet): MR3064670
  • [16] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140.
    Mathematical Reviews (MathSciNet): MR2091358
    Zentralblatt MATH: 1058.60037
    Digital Object Identifier: doi:10.1016/j.jfa.2004.01.002
  • [17] Gubinelli, M. (2010). Ramification of rough paths. J. Differential Equations 248 693–721.
    Mathematical Reviews (MathSciNet): MR2578445
    Zentralblatt MATH: 05671636
    Digital Object Identifier: doi:10.1016/j.jde.2009.11.015
  • [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.
    Mathematical Reviews (MathSciNet): MR428489
    Zentralblatt MATH: 0364.60097
    Digital Object Identifier: doi:10.2307/3213260
  • [19] Holland, M. and Melbourne, I. (2007). Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. (2) 76 345–364.
    Mathematical Reviews (MathSciNet): MR2363420
    Zentralblatt MATH: 1126.37006
    Digital Object Identifier: doi:10.1112/jlms/jdm060
  • [20] Kelly, D. and Melbourne, I. (2014). Deterministic homogenization for fast slow systems with chaotic noise. Available at arXiv:1409.5748.
    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.
    Mathematical Reviews (MathSciNet): MR3265724
    Zentralblatt MATH: 1305.62323
    Digital Object Identifier: doi:10.1088/0951-7715/27/10/2579
  • [23] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1214374
  • [24] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035–1070.
    Mathematical Reviews (MathSciNet): MR1112406
    Zentralblatt MATH: 0742.60053
    Digital Object Identifier: doi:10.1214/aop/1176990334
    Project Euclid: euclid.aop/1176990334
  • [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.
    Mathematical Reviews (MathSciNet): MR614747
    Zentralblatt MATH: 0458.60074
    Digital Object Identifier: doi:10.1137/0140044
  • [26] Lamperti, J. (1962). On convergence of stochastic processes. Trans. Amer. Math. Soc. 104 430–435.
    Mathematical Reviews (MathSciNet): MR143245
    Zentralblatt MATH: 0113.33502
    Digital Object Identifier: doi:10.1090/S0002-9947-1962-0143245-1
  • [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.
    Mathematical Reviews (MathSciNet): MR2243956
  • [28] Lejay, A. and Victoir, N. (2006). On $(p,q)$-rough paths. J. Differential Equations 225 103–133.
    Mathematical Reviews (MathSciNet): MR2228694
    Zentralblatt MATH: 1097.60048
    Digital Object Identifier: doi:10.1016/j.jde.2006.01.018
  • [29] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
    Mathematical Reviews (MathSciNet): MR1654527
    Zentralblatt MATH: 0923.34056
    Digital Object Identifier: doi:10.4171/RMI/240
  • [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.
    Mathematical Reviews (MathSciNet): MR1829529
    Zentralblatt MATH: 1017.86001
    Digital Object Identifier: doi:10.1002/cpa.1014
  • [31] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620–628.
    Mathematical Reviews (MathSciNet): MR358933
    Digital Object Identifier: doi:10.1214/aop/1176996608
  • [32] Melbourne, I. and Nicol, M. (2009). A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37 478–505.
    Mathematical Reviews (MathSciNet): MR2510014
    Zentralblatt MATH: 1176.37006
    Digital Object Identifier: doi:10.1214/08-AOP410
    Project Euclid: euclid.aop/1241099919
  • [33] Norman, M. F. (1974). Markovian learning processes. SIAM Rev. 16 143–162.
    Mathematical Reviews (MathSciNet): MR343372
    Zentralblatt MATH: 0253.60076
    Digital Object Identifier: doi:10.1137/1016025
  • [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.
    Mathematical Reviews (MathSciNet): MR2415440
    Digital Object Identifier: doi:10.1098/rsta.2008.0051
  • [35] Perkowski, N. and Promel, D. J. (2013). Pathwise stochastic integrals for model free finance. Preprint. Available at arXiv:1311.6187.
    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.
    Mathematical Reviews (MathSciNet): MR1428751
    Zentralblatt MATH: 0876.60015
    Digital Object Identifier: doi:10.1214/aoap/1034625254
    Project Euclid: euclid.aoap/1034625254
  • [37] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.
    Mathematical Reviews (MathSciNet): MR1245252
    Zentralblatt MATH: 0792.60046
    Digital Object Identifier: doi:10.1007/BF01195073
  • [38] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprint of the 1997 edition.
    Mathematical Reviews (MathSciNet): MR2190038
  • [39] Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
    Mathematical Reviews (MathSciNet): MR400329
    Digital Object Identifier: doi:10.1007/BF00532868
  • [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.
    Mathematical Reviews (MathSciNet): MR544194
    Digital Object Identifier: doi:10.2307/1426858
  • [41] Watterson, G. A. (1964). The application of diffusion theory to two population genetic models of Moran. J. Appl. Probab. 1 233–246.
    Mathematical Reviews (MathSciNet): MR168048
    Zentralblatt MATH: 0137.39704
    Digital Object Identifier: doi:10.2307/3211857
  • [42] Young, L.-S. (1998). Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147 585–650.
    Mathematical Reviews (MathSciNet): MR1637655
    Zentralblatt MATH: 0945.37009
    Digital Object Identifier: doi:10.2307/120960
  • [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.
    Mathematical Reviews (MathSciNet): MR2783188
    Zentralblatt MATH: 1236.60069
    Digital Object Identifier: doi:10.1137/090762336

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