Estimates for the probability that Itô processes remain near a path

Let W=(Wi)i[set membership, variant]N be an infinite dimensional Brownian motion and (Xt)t>=0 a continuous adaptedn-dimensional process. Set [tau]R=inf{t:Xt-xt>=Rt}, where xt,t>=0 is a Rn-valued deterministic differentiable curve and Rt>0,t>=0 a time-dependent radius. We assume that, up to [tau]R, the process X solves the following (not necessarily Markov) SDE:. Under local conditions on the coefficients, we obtain lower bounds for P([tau]R>=T) as well as estimates for distribution functions and expectations. These results are discussed in the elliptic and log-normal frameworks. An example of a diffusion process that satisfies the weak Hörmander condition is also given.