Smoothness and Gaussian Density Estimates for Stochastic Functional Differential Equations with Fractional Noise

  • Nguyen Van Tan
DOI: https://doi.org/10.19139/soic-2310-5070-784
Keywords: Stochastic functional differential equations, Density estimates, Malliavin calculus, fractional Brownian motion

Abstract

In this paper, we study the density of the solution to a class of stochastic functional differential equations driven by fractional Brownian motion. Based on the techniques of Malliavin calculus, we prove the smoothness and establish upper and lower Gaussian estimates for the density.

References

F. Baudoin, C. Ouyang, S. Tindel, Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions. Ann. Inst. Henri Poincare Probab. Stat. 50 (2014), no. 1, 111--135.

M. Besalu, A. Kohatsu-Higa, S. Tindel, Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions. Ann. Probab. 44 (2016), no. 1, 399--443.

N.T. Dung, Linear multifractional stochastic Volterra integro-differential equations, Taiwanese J. Math. 17 (1) (2013) 333--350.

N.T. Dung, The density of solutions to multifractional stochastic Volterra integro-differential equations, Nonlinear Anal. 130 (2016), 176-189.

N.T. Dung, Gaussian density estimates for the solution of singular stochastic Riccati equations. Appl. Math. 61 (2016), no. 5, 515-526.

N.T. Dung, N. Privault, G.L. Torrisi, Gaussian estimates for the solutions of some one-dimensional stochastic equations. Potential Anal (2015) 43:289-311.

J. Liu, Ciprian A. Tudor, Analysis of the density of the solution to a semilinear SPDE with fractional noise. Stochastics 88 (2016), no. 7, 959-979.

D. Nualart, The Malliavin calculus and related topics. Probability and its Applications. Springer-Verlag, Berlin, second edition, 2006.

I. Nourdin and F.G. Viens, Density formula and concentration inequalities with Malliavin calculus, Electron. J. Probab., 14:2287-2309, 2009.

A. Takeuchi, Malliavin calculus for degenerate stochastic functional differential equations. Acta Appl. Math. 97 (2007), no. 1-3, 281-295.

M. Zahle, Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields, 111:333-374, 1998.

Published
2020-07-01
How to Cite
Van Tan, N. (2020). Smoothness and Gaussian Density Estimates for Stochastic Functional Differential Equations with Fractional Noise. Statistics, Optimization & Information Computing, 8(4), 822-833. https://doi.org/10.19139/soic-2310-5070-784
Issue
Vol 8 No 4 (2020)
Section
Research Articles
Authors who publish with this journal agree to the following terms:

    1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
    2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
    3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).