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Let Xt be a solution to a stochastic differential equation. Easily verified conditions on the coefficients of the equation give Lp inequalities for stopping times of Xt and the maximal function. An application to Brownian motion with radial drift is also discussed. Introduction. Let B(t) be n-dimensional Brownian motion (n > 1). Denote by Ex the expectation associated with B(t) starting at x. For any stopping time r of B(t) let B(t)* be the maximal function of B up to time r:doi:10.1090/s0002-9947-1986-0833708-1 fatcat:uczxmiuzerb7pp3zsry6vjrmku