# zbMATH — the first resource for mathematics

Trees and asymptotic expansions for fractional stochastic differential equations. (English) Zbl 1172.60017
Summary: 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 $$E[f(X_t)]$$ in terms of $$t$$, where $$X$$ denotes the solution to the SDE and $$f:\mathbb R^n\to\mathbb R$$ is a regular function. Comparing to F. Baudoin and L. Coutin [Stochastic Processes Appl. 117, No. 5, 550–574 (2007; Zbl 1119.60043)], 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 $$H>1/2$$.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G17 Sample path properties 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60G15 Gaussian processes
Full Text:
##### References:
 [1] E. Alòs, O. Mazet and D. Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001) 766-801. · Zbl 1015.60047 · doi:10.1214/aop/1008956692 [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. · Zbl 1119.60043 · doi:10.1016/j.spa.2006.09.004 [3] G. Ben Arous. Flot et séries de Taylor stochastiques. Probab. Theory Related Fields 81 (1989) 29-77. · Zbl 0639.60062 · doi:10.1007/BF00343737 [4] C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984) 191-203. · Zbl 0555.60008 [5] L. Coutin and Z. Qian. Stochastic rough path analysis and fractional Brownian motion. Probab. Theory Related Fields 122 (2002) 108-140. · Zbl 1047.60029 · doi:10.1007/s004400100158 [6] P. E. Kloeden and E. Platen. Numerical Solutions of Stochastic Differential Equations , 3rd edition. Springer, Berlin, 1999. · Zbl 0701.60054 [7] T. Lyons and Z. Qian. System Control and Rough Paths . Oxford Univ. Press, 2002. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001 [8] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555 [9] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002 [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. · Zbl 1144.34038 · doi:10.1007/978-3-540-70847-6_17 [11] A. Neuenkirch. Reconstruction of fractional diffusions. In preparation, 2007. · Zbl 1141.60043 [12] A. Neuenkirch, I. Nourdin and S. Tindel. Delay equations driven by rough paths. Preprint, 2007. · Zbl 1190.60046 [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. · Zbl 1091.60008 · doi:10.1016/j.spl.2005.10.021 [14] I. Nourdin and T. Simon. Correcting Newton-Cotes integrals by Lévy areas. Bernoulli 13 (2007) 695-711. · Zbl 1132.60047 · doi:10.3150/07-BEJ6015 · euclid:bj/1186503483 [15] I. Nourdin and C. A. Tudor. Some linear fractional stochastic equations. Stochastics 78 (2006) 51-65. · Zbl 1102.60050 · doi:10.1080/17442500600688997 [16] D. Nualart. The Malliavin Calculus and Related Topics , 2nd edition. Springer, Berlin, 2006. · Zbl 1099.60003 [17] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. · Zbl 1018.60057 · eudml:42846 [18] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Preprint, Barcelona, 2006. · Zbl 1169.60013 [19] V. Pipiras and M. S. Taqqu. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000) 251-291. · Zbl 0970.60058 · doi:10.1007/s004400000080 [20] E. Platen and W. Wagner. On a Taylor formula for a class of Itô processes. Probab. Math. Statist. 2 (1982) 37-51. · Zbl 0528.60053 [21] A. Rößler. Stochastic Taylor expansions for the expectation of functionals of diffusion processes. Stochastic Anal. Appl. 22 (2004) 1553-1576. · Zbl 1065.60068 · doi:10.1081/SAP-200029495 [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. · Zbl 1094.65008 · doi:10.1080/07362990500397699 [23] A. A. Ruzmaikina. Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000) 1049-1069. · Zbl 0970.60045 · doi:10.1023/A:1018754806993 [24] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. · Zbl 0918.60037 · doi:10.1007/s004400050171
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.