The Best Doob-Type Bounds for the Maximum of Brownian Paths

where Xt= jBtjp and St = max 0 r t jBrjp. In the case p=1 the closed form for s 7! g (s) is found. This yields 1;q = (q(q+1)=2)( (1+(q+1)=q)) for all q > 0 . In the case p 6= 1 no closed form for s 7! g (s) seems to exist. The inequality above holds also in the case p = q+1 (Doob’s maximal inequality ). In this case the equation above (with K = p=2c ) admits g (s) = s as the maximal solution, and the equality is attained only in the limit through the stopping times = (c) when c tends to the best value q+1;q = (q+1) =2q from above. The method of proof relies upon the principle of smooth fit of Kolmogorov and the maximality principle. The results obtained extend to the case when B starts at any given point, as well as to all non-negative submartingales.