Comparison-type theorems for Ito processes and differentially subordinated semimartingales

Let alpha >= 0 and let X, Y be Ito processes dX(t) = phi(t) dB(t) + psi(t) dt, dY(t) = zeta(t) dB(t) + xi(t) dt such that X-0 = x, Y-0 = y, vertical bar phi vertical bar >= vertical bar zeta vertical bar and alpha psi >= vertical bar xi vertical bar. We determine the best universal constant U alpha( x, y) such that P(sup(t)Y(t) >= 0) <= parallel to X+parallel to(1) + U-alpha(x, y). As an application, we compute, for any t is an element of [0, 1] and beta is an element of R, the number L(x, y, t) = inf{parallel to X+parallel to(1) : P(Y* >= beta) >= t}. We also study these problems for a wider class of alpha-subordinated semimartingales and establish a related estimate for smooth functions on Euclidean domains.