TY  - JOUR
TI  - Results on the supremum of fractional Brownian motion
AB  - We show that the distribution of the square of the supremum of reflected fractional Brownian motion up to time a, with Hurst parameter H greater than 1/2, is related to the distribution of its hitting time to level 1, using the self similarity property of fractional Brownian motion. It is also proven that the second moment of supremum of reflected fractional Brownian motion up to time a is bounded above by $a^ {2H}$. Similar relations are obtained for the supremum of fractional Brownian motion with Hurst parameter greater than 1/2, and its hitting time to level 1. What is more, we obtain an upper bound on the complementary probability distribution of the supremum of fractional Brownian motion and re°ected fractional Brownian motion up to time a, using Jensen's and Markov's inequalities. A sharper bound is observed on the distribution of the supremum of fractional Brownian motion by the properties of Gamma distribution. Finally, applications of the given results to financial markets are investigated, and partial results are provided.
AU  - VARDAR ACAR, CEREN
PY  - 2011
JO  - Hacettepe Journal of Mathematics and Statistics
VL  - 40
IS  - 2
SN  - 1303-5010
SP  - 255
EP  - 264
DB  - TRDizin
UR  - http://search/yayin/detay/121521
ER  -