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Let B=(B t ) t≥0 be a standard Brownian motion. For c>0, k>0, let T(c, k)=inf{t≥0 : max s≤t B s −cB t ≥k}, T*(c, k)=inf{t≥0 : max s≤t |B s |−c|B t |≥k}. We show that for the c>0 and k>0, both T(c, k) and T*(c, k) are finite almost everywhere. Moreover T(c, k) and T*(c, k)∈L p/2 if and only of c 1, and for all c>0 when p≤1. These results have analogues for simple random walks

Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion