Maximal Inequalities for Fractional Brownian Motion with Variable Drift
Main Article Content
Abstract
Let BH be a fractional Brownian motion with H∈ (0, 1) and g be a deterministic function. We study the asymptotic behaviour of the tail probability as for fixed x and as for fixed T. Our results partially generalise those obtained by Prakasa Rao in [1].
Keywords:
Fractional Brownian motion, Maximal inequalities, Variable drift.
References
[1] B. L. S. Prakasa Rao: Some maximal inequalities for fractional Brownian motion with polynomial drift. Stoch. Anal. Appl. 31, no. 5, (2013) 785-799. https://doi.org/10.1080/07362994.2013.817240.
[2] B. L. S Prakasa Rao: Statistical Inference for Fractional Diffusion Processes. Wiley, Chichester, UK (2010).
[3] L. Jiao: Some limit results for probabilities estimates of Brownian motion with polynomial drift. Indian J. Pure Appl. Math. 41, no. 3, (2010) 425-442. https://doi.org/10.1007/s13226-010-0026-9.
[4] D. Nualart: The Malliavin calculus and related topics. Probability and its Applications. Springer¬Verlag, Berlin, second edition (2006).
[5] N. Privault: Stochastic analysis in discrete and continuous settings with normal martingales. Lecture Notes in Mathematics, 1982. Springer-Verlag, Berlin, 2009.
[6] E. Alòs, J. A. León, D. Nualart: Stochastic Stratonovich calculus fBm for fractional Brownian motion with Hurst parameter less than 1. Taiwanese J. Math. 5, no. 3, (2001) 609-632.
[7] I. Nourdin: A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. Séminaire de probabilities XLI, 181-197, Lecture Notes in Math., 1934, Springer, Berlin (2008) 181-197. https://doi.org/10.1007/978-3-540-77913-1-8.
[2] B. L. S Prakasa Rao: Statistical Inference for Fractional Diffusion Processes. Wiley, Chichester, UK (2010).
[3] L. Jiao: Some limit results for probabilities estimates of Brownian motion with polynomial drift. Indian J. Pure Appl. Math. 41, no. 3, (2010) 425-442. https://doi.org/10.1007/s13226-010-0026-9.
[4] D. Nualart: The Malliavin calculus and related topics. Probability and its Applications. Springer¬Verlag, Berlin, second edition (2006).
[5] N. Privault: Stochastic analysis in discrete and continuous settings with normal martingales. Lecture Notes in Mathematics, 1982. Springer-Verlag, Berlin, 2009.
[6] E. Alòs, J. A. León, D. Nualart: Stochastic Stratonovich calculus fBm for fractional Brownian motion with Hurst parameter less than 1. Taiwanese J. Math. 5, no. 3, (2001) 609-632.
[7] I. Nourdin: A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. Séminaire de probabilities XLI, 181-197, Lecture Notes in Math., 1934, Springer, Berlin (2008) 181-197. https://doi.org/10.1007/978-3-540-77913-1-8.