One-sided maximal inequalities for a stock process

Let X=(Xt)t≥0 be a stock process, assumed a geometric Brownian motion, with drift μ>0 and volatility σ>0 starting at x>0. For any stopping time τ for X, we establish an upper moment bound for Ex(max0≤t≤τ⁡Xt) under certain restrictions. 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.