Estimates for the probability that Itô processes remain near a path

AbstractLet W=(Wi)i∈N be an infinite dimensional Brownian motion and (Xt)t≥0 a continuous adaptedn-dimensional process. Set τR=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 τR, the process X solves the following (not necessarily Markov) SDE:Xt∧τR=x+∑j=1∞∫0t∧τRσj(s,ω,Xs)dWsj+∫0t∧τRb(s,ω,Xs)ds. Under local conditions on the coefficients, we obtain lower bounds for P(τR≥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