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Estimates for the probability that Ito processes remain near a path

Abstract : Let W = (W(i))(i is an element of N) he an infinite dimensional Brownian motion and (X(t))(1 >= 0) a continuous adapted n-dimensional process. Set tau(R) = infit : {X(t) - x(t)vertical bar}, 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 tau(R), the process X solves the following (not necessarily Markov) SDE : X(t Lambda tau R) = x + Sigma(infinity)(j=1) integral(t Lambda tau R)(0) sigma(j) (s, omega, X(s))dW(s)(j) + integral(t Lambda tau R)(0) b(s, omega, X(s))ds. 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 Hormander condition is also given. (C) 2011 Elsevier B.V. All rights reserved.
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Vlad Bally, Begona Fernandez, Ana Meda. Estimates for the probability that Ito processes remain near a path. Stochastic Processes and their Applications, Elsevier, 2011, 121 (9), pp.2087--2113. ⟨10.1016/⟩. ⟨hal-00692821⟩



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