Estimates for the probability that Itô processes remain near a path

Let W = ( W i ) i ∈ N be an infinite dimensional Brownian motion and ( X t ) t ≥ 0 a continuous adapted n -dimensional process. Set τ R = inf { t : | X t − x t | ≥ R t } , where x t , t ≥ 0 is a R n -valued deterministic differentiable curve and R t > 0 , t ≥ 0 a time-dependent radius. We assume that, up to τ R , the process X solves the following (not necessarily Markov) S D E : X t ∧ τ R = x + ∑ j = 1 ∞ ∫ 0 t ∧ τ R σ j ( s , ω , X s ) d W s j + ∫ 0 t ∧ τ R b ( s , ω , X s ) d s . Under local conditions on the coefficients, we obtain lower bounds for P ( τ 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.