Electronic Journal of Probability

Short time kernel asymptotics for rough differential equation driven by fractional Brownian motion

Yuzuru Inahama

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Abstract

We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter $H ~(1/3 < H \le 1/2)$ under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe’s distributional Malliavin calculus.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 34, 29 pp.

Dates
Received: 27 February 2015
Accepted: 28 March 2016
First available in Project Euclid: 22 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1461332875

Digital Object Identifier
doi:10.1214/16-EJP4144

Mathematical Reviews number (MathSciNet)
MR3492938

Zentralblatt MATH identifier
1338.60142

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60F99: None of the above, but in this section 60G22: Fractional processes, including fractional Brownian motion

Keywords
rough path theory Malliavin calculus fractional Brownian motion short time asymptotic expansion

Rights
Creative Commons Attribution 4.0 International License.

Citation

Inahama, Yuzuru. Short time kernel asymptotics for rough differential equation driven by fractional Brownian motion. Electron. J. Probab. 21 (2016), paper no. 34, 29 pp. doi:10.1214/16-EJP4144. https://projecteuclid.org/euclid.ejp/1461332875


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References

  • [1] Aida, S.: Vanishing of one-dimensional $L^2$-cohomologies of loop groups. J. Funct. Anal. 261 (2011), no. 8, 2164–2213.
  • [2] Ben Arous, G.: Noyau de la chaleur hypoelliptique et géométrie sous-riemannienne. Stochastic analysis (Paris, 1987), 1–16, Lecture Notes in Math., 1322, Springer, Berlin, 1988.
  • [3] Ben Arous, G.: Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus. Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 307–331.
  • [4] Ben Arous, G.: Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann. Inst. Fourier (Grenoble) 39 (1989), no. 1, 73–99.
  • [5] Ben Arous, G. and Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale. I. Probab. Theory Related Fields 90 (1991), no. 2, 175–202.
  • [6] Ben Arous, G. and Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale. II. Probab. Theory Related Fields 90 (1991), no. 3, 377–402.
  • [7] Azencott, R.: Densité des diffusions en temps petit: développements asymptotiques. I. Seminar on probability, XVIII, 402–498, Lecture Notes in Math., 1059, Springer, Berlin, 1984.
  • [8] Bailleul, I.: Flows driven by rough paths. Rev. Mat. Iberoam. 31 (2015), no. 3, 901–934.
  • [9] Bailleul, I.: Regularity of the Ito-Lyons map. Confluentes Math. 7 (2015), no. 1, 3–11.
  • [10] Baudoin, F., Nualart, E., Ouyang, C. and Tindel, S.: On probability laws of solutions to differential systems driven by a fractional Brownian motion. To appear in Ann. Probab. arXiv:1401.3583
  • [11] Baudoin, F. and Ouyang, C.: On small time asymptotics for rough differential equations driven by fractional Brownian motions. In Large deviations and asymptotic methods in finance, 413–438. Springer, 2015.
  • [12] Baudoin, F., Ouyang, C. and Zhang, X.: Varadhan Estimates for rough differential equations driven by fractional Brownian motions. Stochastic Process. Appl. 125 (2015), Issue 2, 634–652.
  • [13] Baudoin, F., Ouyang, C. and Zhang, X.: Smoothing effect of rough differential equations driven by fractional Brownian motions. Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 1, 412–428.
  • [14] Bismut, J. M.: Large deviations and the Malliavin calculus. Progress in Mathematics, 45. Birkhäuser Boston, Inc., Boston, MA, 1984.
  • [15] Boedihardjo, H., Geng, X. and Qian, Z.: Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities. To appear in Osaka J. Math. arXiv:1309.0835
  • [16] Cass, T. and Friz, P.: Densities for rough differential equations under Hörmander’s condition. Ann. of Math. (2) 171 (2010), no. 3, 2115–2141.
  • [17] Cass, T., Friz, P. and Victoir, N.: Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009), no. 6, 3359–3371.
  • [18] Cass, T., Hairer, M., Litterer, C. and Tindel, S.: Smoothness of the density for solutions to Gaussian Rough Differential Equations. Ann. Probab. 43 (2015), no. 1, 188–239.
  • [19] Cass, T., Litterer, C. and Lyons, T.: Integrability and tail estimates for Gaussian rough differential equations. Ann. Probab. 41 (2013), no. 4, 3026–3050.
  • [20] Driscoll, P.: Smoothness of densities for area-like processes of fractional Brownian motion. Probab. Theory Related Fields 155 (2013), no. 1–2, 1–34.
  • [21] Friz, P., Gess. B., Gulisashvili, A., and Riedel, S.: Jain-Monrad criterion for rough paths and applications. Ann. Probab. 44 (2016), no. 1, 684–738.
  • [22] Friz, P. and Oberhauser, H.: A generalized Fernique theorem and applications. Proc. Amer. Math. Soc. 138 (2010), no. 10, 3679–3688.
  • [23] Friz, P. and Victoir, N.: A variation embedding theorem and applications. J. Funct. Anal. 239 (2006), no. 2, 631–637.
  • [24] Friz, P. and Victoir, N.: Large deviation principle for enhanced Gaussian processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 6, 775–785.
  • [25] Friz, P. and Victoir, N.: Multidimensional stochastic processes as rough paths. Cambridge University Press, Cambridge, 2010.
  • [26] Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139 (1977), no. 1–2, 95–153.
  • [27] Hairer, M. and Pillai, N.: Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41 (2013), no. 4, 2544–2598
  • [28] Hu, Y. and Tindel, S.: Smooth Density for Some Nilpotent Rough Differential Equations. J. Theoret. Probab. 26 (2013), no. 3, 722–749.
  • [29] Ikeda, N. and Watanabe, S.: Stochastic differential equations and diffusion processes. Second edition. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
  • [30] Inahama, Y.: Quasi-sure existence of Brownian rough paths and a construction of Brownian pants. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 4, 513–528.
  • [31] Inahama, Y.: A stochastic Taylor-like expansion in the rough path theory. J. Theoret. Probab. 23 (2010) 671–714.
  • [32] Inahama, Y.: Laplace approximation for rough differential equation driven by fractional Brownian motion, Ann. Probab. 41 (2013), No. 1, 170–205.
  • [33] Inahama, Y.: Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory. J. Math. Soc. Japan 68, No. 2 (2016), 1–43. arXiv:1110.2604
  • [34] Inahama, Y.: Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes. Trans. Amer. Math. Soc. 367 (2015), 8107–8137.
  • [35] Inahama, Y.: Malliavin differentiability of solutions of rough differential equations. J. Funct. Anal. 267 (2014), 1566–1584.
  • [36] Inahama, Y. and Kawabi, H.: Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths. J. Funct. Anal. 243 (2007), no. 1, 270–322.
  • [37] Kuo, H. H.: Gaussian measures in Banach spaces. Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975.
  • [38] Kusuoka, S. and Stroock, D. W.: Precise asymptotics of certain Wiener functionals. J. Funct. Anal. 99 (1991), no. 1, 1–74.
  • [39] Kusuoka, S. and Stroock, D. W.: Asymptotics of certain Wiener functionals with degenerate extrema. Comm. Pure Appl. Math. 47 (1994), no. 4, 477–501.
  • [40] Léandre, R.: Majoration en temps petit de la densité d’une diffusion dégénérée. Probab. Theory Related Fields 74 (1987), no. 2, 289–294.
  • [41] Léandre, R.: Minoration en temps petit de la densité d’une diffusion dégénérée. J. Funct. Anal. 74 (1987), no. 2, 399–414.
  • [42] Léandre, R.: Intégration dans la fibre associée à une diffusion dégénérée. Probab. Theory Related Fields 76 (1987), no. 3, 341–358.
  • [43] Léandre, R.: Applications quantitatives et géométriques du calcul de Malliavin. Stochastic analysis (Paris, 1987), 109–133, Lecture Notes in Math., 1322, Springer, Berlin, 1988.
  • [44] Léandre, R.: Développement asymptotique de la densité d’une diffusion dégénérée. Forum Math. 4 (1992), no. 1, 45–75.
  • [45] Lejay, A.: An introduction to rough paths. Séminaire de Probabilités XXXVII, 1–59, Lecture Notes in Math., 1832, Springer, Berlin, 2003.
  • [46] Lyons, T., Caruana, M. and Lévy, T.: Differential equations driven by rough paths. Lecture Notes in Math., 1908, Springer, Berlin, 2007.
  • [47] Lyons, T. and Qian, Z.: System control and rough paths. Oxford University Press, Oxford, 2002.
  • [48] Molčanov, S. A.: Diffusion processes, and Riemannian geometry. Uspehi Mat. Nauk 30 (1975), no. 1 (181), 3–59. (English translation: Russian Math. Surveys 30 (1975), no. 1, 1–63.)
  • [49] Nualart, D.: The Malliavin calculus and related topics. Second edition. Springer-Verlag, Berlin, 2006.
  • [50] Shigekawa, I.: Stochastic analysis. Translations of Mathematical Monographs, 224. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2004.
  • [51] Takanobu, S.: Diagonal short time asymptotics of heat kernels for certain degenerate second order differential operators of Hörmander type. Publ. Res. Inst. Math. Sci. 24 (1988), no. 2, 169–203.
  • [52] Takanobu, S. and Watanabe, S.: Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations. Asymptotic problems in probability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990), 194–241, Pitman Res. Notes Math. Ser., 284, Longman Sci. Tech., Harlow, 1993.
  • [53] Uemura, H.: On a short time expansion of the fundamental solution of heat equations by the method of Wiener functionals. J. Math. Kyoto Univ. 27 (1987), no. 3, 417–431.
  • [54] Uemura, H.: Off-diagonal short time expansion of the heat kernel on a certain nilpotent Lie group. J. Math. Kyoto Univ. 30 (1990), no. 3, 403–449.
  • [55] Uemura, H. and Watanabe, S.: Diffusion processes and heat kernels on certain nilpotent groups. Stochastic analysis (Paris, 1987), 173–197, Lecture Notes in Math., 1322, Springer, Berlin, 1988.
  • [56] Watanabe, S.: Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15 (1987), no. 1, 1–39.