Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 50, Issue 1, 2014, Pages 111-135

  Baudoin, Fabrice ^a   Ouyang, Cheng ^b   Tindel, Samy ^c  

^a PURDUE UNIVERSITY   (United States)

^b UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^c UNIVERSITÉ DE LORRAINE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84891911724     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1214/12-AIHP522     Document Type: Article

References (36)

1. F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007) 550-574. MR2320949
2. F. Baudoin and M. Hairer. A version of Hörmander's theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 (2007) 373-395. MR2322701
3. F. Baudoin and C. Ouyang. Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions. Stochastic. Process. Appl. 121 (2011) 759-792. MR2770906
4. F. Baudoin, E. Nualart, C. Ouyang and S. Tindel. Work in progress. Preprint, 2012.
5. M. Besalú and D. Nualart. Estimates for the solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈ (13, 12) . Stoch. Dyn. 11 (2011) 243-263. MR2836524
6. M. Capitaine, E. Hsu and M. Ledoux. Martingale representation and logarithmic Sobolev inequality. Electron. Com. Probab. 2 (1997) 71-81. MR1484557
7. T. Cass and P. Friz. Densities for rough differential equations under Hörmander condition. Ann. Math. To appear.
8. T. Cass, P. Friz and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009) 3359-3371. MR2485431
9. T. Cass, C. Litterer and T. Lyons. Integrability estimates for Gaussian rough differential equations. Arxiv preprint, 2011.
10. A. Chronopoulou and S. Tindel. On inference for fractional differential equations. Arxiv preprint, 2011.
11. L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108-140. MR1883719
12. R. C. Dalang and E. Nualart. Potential theory for hyperbolic SPDEs. Ann. Probab. 32(3A) (2004) 2099-2148. MR2073187
13. A. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. 2007 (2007) abm009. MR2387018
14. A. Deya, A. Neuenkirch and S. Tindel. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 48(2) (2012) 518-550.
15. P. Driscoll. Smoothness of density for the area process of fractional Brownian motion. Arxiv preprint, 2010.
16. P. Friz and N. Victoir. Multidimensional Stochastic Processes Seen as Rough Paths. Cambridge Univ. Press, Cambridge, 2010. MR2604669
17. M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. MR2091358
18. M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38 (2010) 1-75. MR2599193
19. M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownianmotion. Ann. Probab. 33 (2005) 703-758. MR2123208
20. M. Hairer and A. Ohashi. Ergodicity theory of SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950-1977. MR2349580
21. M. Hairer and N. S. Pillai. Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Preprint, 2011.
22. E. Hsu. Stochastic Analysis on Manifolds. Graduate Series in Mathematics 38. Amer. Math. Soc., Providence, RI, 2002. MR1882015
23. Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functions of order greater than 1/2. Abel Symp. 2 (2007) 349-413. MR2397797
24. S. Kou and X. Sunney-Xie. Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule. Phys. Rev. Lett. 93 (2004) 18.
25. M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001. MR1849347
26. T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, Oxford, 2002. MR2036784
27. A. Millet and M. Sanz-Solé. Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. 42 (2006) 245-271. MR2199801
28. A. Neuenkirch, I. Nourdin, A. Rößler and S. Tindel. Trees and asymptotic developments for fractional diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 157-174. MR2500233
29. A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the Lévy area. Stochastic Process. Appl. 120 (2010) 223-254. MR2576888
30. D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer-Verlag, Berlin, 2006. MR2200233
31. D. Nualart and A. Rǎ;̧scanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. MR1893308
32. D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 (2009) 391-409. MR2493996
33. J. Szymanski and M. Weiss. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103 (2009) 3.
34. V. Tejedor, O. Bénichou, R. Voituriez, R. Jungmann, F. Simmel, C. Selhuber-Unkel, L. Oddershede and R. Metzle. Quantitative analysis of single particle trajectories: Mean maximal excursion method. Biophysical J. 98 (2010) 1364-1372.
35. A. S. Üstünel. Analysis on Wiener space and applications. Arxiv preprint, 2010.
36. M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. MR1640795