The Annals of Probability

Laplace approximation for rough differential equation driven by fractional Brownian motion

Yuzuru Inahama

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We consider a rough differential equation indexed by a small parameter $\varepsilon>0$. When the rough differential equation is driven by fractional Brownian motion with Hurst parameter $H$ ($1/4<H<1/2$), we prove the Laplace-type asymptotics for the solution as the parameter $\varepsilon$ tends to zero.

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Ann. Probab., Volume 41, Number 1 (2013), 170-205.

First available in Project Euclid: 23 January 2013

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Primary: 60G22: Fractional processes, including fractional Brownian motion 60F99: None of the above, but in this section 60H10: Stochastic ordinary differential equations [See also 34F05]

Rough path theory Laplace approximation fractional Brownian motion


Inahama, Yuzuru. Laplace approximation for rough differential equation driven by fractional Brownian motion. Ann. Probab. 41 (2013), no. 1, 170--205. doi:10.1214/11-AOP733.

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  • [1] Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York–London.
  • [2] Aida, S. (2007). Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold. J. Funct. Anal. 251 59–121.
  • [3] Albeverio, S., Röckle, H. and Steblovskaya, V. (2000). Asymptotic expansions for Ornstein–Uhlenbeck semigroups perturbed by potentials over Banach spaces. Stochastics Stochastics Rep. 69 195–238.
  • [4] Azencott, R. (1982). Formule de Taylor stochastique et développement asymptotique d’intégrales de Feynman. In Seminar on Probability, XVI, Supplement. Lecture Notes in Math. 921 237–285. Springer, Berlin.
  • [5] Baudoin, F. and Coutin, L. (2007). Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 550–574.
  • [6] Baudoin, F. and Coutin, L. (2008). Self-similarity and fractional Brownian motions on Lie groups. Electron. J. Probab. 13 1120–1139.
  • [7] Ben Arous, G. (1988). Methods de Laplace et de la phase stationnaire sur l’espace de Wiener. Stochastics 25 125–153.
  • [8] Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.
  • [9] Coutin, L. (2007). An introduction to (stochastic) calculus with respect to fractional Brownian motion. In Séminaire de Probabilités XL. Lecture Notes in Math. 1899 3–65. Springer, Berlin.
  • [10] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108–140.
  • [11] Eldredge, N. (2005). Computing $p$-variation. Unpublished manuscript. Univ. California, San Diego.
  • [12] Friz, P. and Oberhauser, H. (2010). A generalized Fernique theorem and applications. Proc. Amer. Math. Soc. 138 3679–3688.
  • [13] Friz, P. and Victoir, N. (2006). A variation embedding theorem and applications. J. Funct. Anal. 239 631–637.
  • [14] Friz, P. and Victoir, N. (2007). Large deviation principle for enhanced Gaussian processes. Ann. l’Inst. Henri Poincaré Probab. Stat. 43 775–785.
  • [15] Friz, P. and Victoir, N. (2010). Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 369–413.
  • [16] 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.
  • [17] Gubinelli, M. and Lejay, A. (2009). Global existence for rough differential equations under linear growth condition. Preprint.
  • [18] Inahama, Y. (2006). Laplace’s method for the laws of heat processes on loop spaces. J. Funct. Anal. 232 148–194.
  • [19] Inahama, Y. (2010). A stochastic Taylor-like expansion in the rough path theory. J. Theoret. Probab. 23 671–714.
  • [20] Inahama, Y. and Kawabi, H. (2007). Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths. J. Funct. Anal. 243 270–322.
  • [21] Jain, N. C. and Monrad, D. (1983). Gaussian measures in $B_{p}$. Ann. Probab. 11 46–57.
  • [22] Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129. Cambridge Univ. Press, Cambridge.
  • [23] Kuo, H. H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 463. Springer, Berlin.
  • [24] Kusuoka, S. and Osajima, Y. (2008). A remark on the asymptotic expansion of density function of Wiener functionals. J. Funct. Anal. 255 2545–2562.
  • [25] Kusuoka, S. and Stroock, D. W. (1991). Precise asymptotics of certain Wiener functionals. J. Funct. Anal. 99 1–74.
  • [26] Kusuoka, S. and Stroock, D. W. (1994). Asymptotics of certain Wiener functionals with degenerate extrema. Comm. Pure Appl. Math. 47 477–501.
  • [27] Lejay, A. (2003). An introduction to rough paths. In Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 1–59. Springer, Berlin.
  • [28] Li, X.-D. and Lyons, T. J. (2006). Smoothness of Itô maps and diffusion processes on path spaces. I. Ann. Sci. Éc. Norm. Supér. (4) 39 649–677.
  • [29] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Univ. Press, Oxford.
  • [30] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [31] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [32] Millet, A. and Sanz-Solé, M. (2006). Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 42 245–271.
  • [33] Mishura, Y. S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math. 1929. Springer, Berlin.
  • [34] Neuenkirch, A., Nourdin, I., Rößler, A. and Tindel, S. (2009). Trees and asymptotic expansions for fractional stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 45 157–174.
  • [35] Rovira, C. and Tindel, S. (2000). Sharp Laplace asymptotics for a parabolic SPDE. Stochastics Stochastics Rep. 69 11–30.
  • [36] Takanobu, S. and Watanabe, S. (1993). Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations. In Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990). Pitman Res. Notes Math. Ser. 284 194–241. Longman Sci. Tech., Harlow.
  • [37] Taniguchi, S. (2008). Quadratic Wiener functionals of square norms on measure spaces. Commun. Stoch. Anal. 2 11–26.