One-sided maximal inequalities for a stock process

Let X = (X-t)(t >= 0) be a stock process, assumed a geometric Brownian motion, with drift mu > 0 and volatility sigma > 0 starting at x > 0. For any stopping time tau for X, we establish an upper moment bound for E-x (max(0 <= t <=tau) X-t) under certain restrictions. o<t<T Our method of proof employs the Gronwall inequality, and a comparison principle for a system of first -order nonlinear differential equations. The bound obtained in this paper extends an existing result, and the method of proof employed is quite new. (C) 2017 Elsevier Inc. All rights reserved.