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. | VNU Journal of Science Mathematics Physics Vol. 36 No. 3 2020 1-9 Review Article Maximal Inequalities for Fractional Brownian Motion with Variable Drift Trinh Nhu Quynh1 Tran Manh Cuong2 1 Military Information Technology Institute 17 Hoang Sam Cau Giay Hanoi Vietnam 2 Department of Mathematics VNU University of Science 334 Nguyen Trai Thanh Xuan Hanoi Vietnam Received 18 December 2019 Revised 06 March 2020 Accepted 15 June 2020 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. 1. Introduction Let B H BtH t 0 be a standard fractional Brownian motion fBm with Hurst index . BH is a centered Gaussian process with covariance function given by 1 2H RH t s E BtH BsH t s 2 H t s t s 0. 2H 2 1 We refer the readers to the monograph 2 for a short survey of properties of fBm. When H 2 the following limit theorems were proved by Prakasa Rao in 1 . Theorem . Let g t ak t ak 1t . a1t be a polynomial of degree k with ak gt 0. Then for k k 1 any T gt 0 and k 2 we have ____ Corresponding author. Email address cuongtm@ https 2588-1124 1 2 . Quynh . Cuong VNU Journal of Science Mathematics Physics Vol. 36 No. 3 2020 1-9 log P sup BtH g t x t 0 T 1 lim 2 . x x 2T 2 H Theorem . Let g t ak t ak 1t k 1 k . a1t be a polynomial of degree k with ak gt 0. Then for any x gt 0 and k 1 we have log P sup BtH g t x t 0 T ak2 lim sup . T T 2k 2 H 2 1 BtH reduces to a standard Brownian motion. In this case Prakasa Rao s It is known that when H 2 results reduce to those established previously by Jiao 3 . Naturally one would like to ask the following questions 1 Q1 Are Theorems and still true when H lt 2 Q2 Can we remove the polynomial structure of the drift g
