Trees and asymptotic expansions for fractional stochastic differential equations
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 157-174.

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.

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.

DOI : 10.1214/07-AIHP159
Classification : 60H05, 60H07, 60G15
Mots clés : fractional brownian motion, stochastic differential equations, trees expansions
@article{AIHPB_2009__45_1_157_0,
     author = {Neuenkirch, A. and Nourdin, I. and R\"o{\ss}ler, A. and Tindel, S.},
     title = {Trees and asymptotic expansions for fractional stochastic differential equations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {157--174},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/07-AIHP159},
     mrnumber = {2500233},
     zbl = {1172.60017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/07-AIHP159/}
}
TY  - JOUR
AU  - Neuenkirch, A.
AU  - Nourdin, I.
AU  - Rößler, A.
AU  - Tindel, S.
TI  - Trees and asymptotic expansions for fractional stochastic differential equations
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 157
EP  - 174
VL  - 45
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/07-AIHP159/
DO  - 10.1214/07-AIHP159
LA  - en
ID  - AIHPB_2009__45_1_157_0
ER  - 
%0 Journal Article
%A Neuenkirch, A.
%A Nourdin, I.
%A Rößler, A.
%A Tindel, S.
%T Trees and asymptotic expansions for fractional stochastic differential equations
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 157-174
%V 45
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/07-AIHP159/
%R 10.1214/07-AIHP159
%G en
%F AIHPB_2009__45_1_157_0
Neuenkirch, A.; Nourdin, I.; Rößler, A.; Tindel, S. Trees and asymptotic expansions for fractional stochastic differential equations. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 157-174. doi : 10.1214/07-AIHP159. http://www.numdam.org/articles/10.1214/07-AIHP159/

[1] E. Alòs, O. Mazet and D. Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001) 766-801. | MR | Zbl

[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. | MR | Zbl

[3] G. Ben Arous. Flot et séries de Taylor stochastiques. Probab. Theory Related Fields 81 (1989) 29-77. | MR | Zbl

[4] C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984) 191-203. | MR | Zbl

[5] L. Coutin and Z. Qian. Stochastic rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 (2002) 108-140. | MR | Zbl

[6] P. E. Kloeden and E. Platen. Numerical Solutions of Stochastic Differential Equations, 3rd edition. Springer, Berlin, 1999. | MR | Zbl

[7] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, 2002. | MR | Zbl

[8] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. | MR | Zbl

[9] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[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. | MR | Zbl

[14] I. Nourdin and T. Simon. Correcting Newton-Cotes integrals by Lévy areas. Bernoulli 13 (2007) 695-711. | MR | Zbl

[15] I. Nourdin and C. A. Tudor. Some linear fractional stochastic equations. Stochastics 78 (2006) 51-65. | MR | Zbl

[16] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006. | MR | Zbl

[17] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. | MR | Zbl

[18] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Preprint, Barcelona, 2006. | MR

[19] V. Pipiras and M. S. Taqqu. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000) 251-291. | MR | Zbl

[20] E. Platen and W. Wagner. On a Taylor formula for a class of Itô processes. Probab. Math. Statist. 2 (1982) 37-51. | MR | Zbl

[21] A. Rößler. Stochastic Taylor expansions for the expectation of functionals of diffusion processes. Stochastic Anal. Appl. 22 (2004) 1553-1576. | MR | Zbl

[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. | MR | Zbl

[23] A. A. Ruzmaikina. Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000) 1049-1069. | MR | Zbl

[24] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. | MR | Zbl

Cité par Sources :