Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Trees and asymptotic expansions for fractional stochastic differential equations

A. Neuenkirch, I. Nourdin, A. Rößler, and S. Tindel

Full-text: Open access

Abstract

In this article, we consider an n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H>1/3. We derive an expansion for E[f(Xt)] in terms of t, where X denotes the solution to the SDE and f:ℝn→ℝ is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl. 117 (2007) 550–574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case H>1/2.

Résumé

Dans cet article, nous considérons une équation différentielle stochastique multidimensionnelle dirigée par un mouvement brownien fractionnaire d’indice de Hurst H>1/3. Nous développons E[f(Xt)] par rapport à t, où on note X la solution de l’EDS et où f:ℝn→ℝ est une fonction régulière. Par rapport à F. Baudoin et L. Coutin, Stochastic Process. Appl. 117 (2007) 550–574, où le même problème est étudié, nous améliorons leur résultat dans trois directions différentes: nous traîtons le cas d’une équation avec dérive, nous paramétrons notre développement à l’aide d’arbres, ce qui le rend plus facile à utiliser, et nous proposons un contrôle plus fin du reste quand H>1/2.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 157-174.

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469976

Digital Object Identifier
doi:10.1214/07-AIHP159

Mathematical Reviews number (MathSciNet)
MR2500233

Zentralblatt MATH identifier
1172.60017

Subjects
Primary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60G15: Gaussian processes

Keywords
Fractional Brownian motion Stochastic differential equations Trees expansions

Citation

Neuenkirch, A.; Nourdin, I.; Rößler, A.; Tindel, S. Trees and asymptotic expansions for fractional stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 157--174. doi:10.1214/07-AIHP159. https://projecteuclid.org/euclid.aihp/1234469976


Export citation

References

  • [1] E. Alòs, O. Mazet and D. Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001) 766–801.
  • [2] F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic. Process. Appl. 117 (2007) 550–574.
  • [3] G. Ben Arous. Flot et séries de Taylor stochastiques. Probab. Theory Related Fields 81 (1989) 29–77.
  • [4] C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984) 191–203.
  • [5] L. Coutin and Z. Qian. Stochastic rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 (2002) 108–140.
  • [6] P. E. Kloeden and E. Platen. Numerical Solutions of Stochastic Differential Equations, 3rd edition. Springer, Berlin, 1999.
  • [7] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, 2002.
  • [8] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310.
  • [9] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86–140.
  • [10] Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functions of order greater than 1/2. Proceedings of Abel Symposium. To appear, 2007.
  • [11] A. Neuenkirch. Reconstruction of fractional diffusions. In preparation, 2007.
  • [12] A. Neuenkirch, I. Nourdin and S. Tindel. Delay equations driven by rough paths. Preprint, 2007.
  • [13] I. Nourdin and T. Simon. On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Statist. Probab. Lett. 76 (2006) 907–912.
  • [14] I. Nourdin and T. Simon. Correcting Newton–Cotes integrals by Lévy areas. Bernoulli 13 (2007) 695–711.
  • [15] I. Nourdin and C. A. Tudor. Some linear fractional stochastic equations. Stochastics 78 (2006) 51–65.
  • [16] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006.
  • [17] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55–81.
  • [18] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Preprint, Barcelona, 2006.
  • [19] V. Pipiras and M. S. Taqqu. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000) 251–291.
  • [20] E. Platen and W. Wagner. On a Taylor formula for a class of Itô processes. Probab. Math. Statist. 2 (1982) 37–51.
  • [21] A. Rößler. Stochastic Taylor expansions for the expectation of functionals of diffusion processes. Stochastic Anal. Appl. 22 (2004) 1553–1576.
  • [22] A. Rößler. Rooted tree analysis for order conditions of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations. Stochastic Anal. Appl. 24 (2006) 97–134.
  • [23] A. A. Ruzmaikina. Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000) 1049–1069.
  • [24] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333–374.