SHARP MAXIMAL INEQUALITY FOR MARTINGALES AND STOCHASTIC INTEGRALS

Let X = (X-t)(t >= 0) be a martingale and H = (H-t)(t >= 0) be a predictable process taking values in [-1,1]. Let Y denote the stochastic integral of H with respect to X. We show that parallel to sup(t >= 0) Y-t parallel to(1) <= beta(0)parallel to sup(t >= 0)vertical bar X-t parallel to vertical bar(1), where beta(0) = 2,0856 ... is the best possible. Furthermore, if, in addition, X is nonnegative, then parallel to sup(t >= 0) Y-t parallel to(1) <= beta(+)(0)parallel to sup(t >= 0)X(t)parallel to(1), where beta(+)(0) = 14/9 is the best possible.