Moment bounds for self-normalized martingales

Let tau be any stopping time of Brownian motion w(t). Recently, Graversen and Peskir (1998) derived an upper bound for the mean of the supremum of the self-normalized process (\w(t)\/root1+t) over 0 less than or equal to t less than or equal to tau. At the cost of adding a universal constant, we extend their result by applying pth powers and exponential functions to it, as well as by considering more general processes, including martingales.