Pathwise uniqueness for a SDE with non-Lipschitz coefficients

AbstractWe consider the ordinary stochastic differential equation dX=−cXdt+2(1−|X|2)dB on the closed unit ball E in Rn. While it is easy to prove existence and distribution uniqueness for solutions of this SDE for each c⩾0, pathwise uniqueness can be proved by standard methods only in dimension n=1 and in dimensions n⩾2 if c=0 or if c⩾2 and the initial condition is in the interior of E. We sharpen these results by proving pathwise uniqueness for c⩾1. More precisely, we show that for X1,X2 solutions relative to the same Brownian motion, the function t↦|X1(t)−X2(t)|2+|1−|X1(t)|2−1−|X2(t)|2|2 is almost surely nonincreasing. Whether or not pathwise uniqueness holds in dimensions n⩾2 for 0<c