Estimates for the probability that Ito processes remain near a path

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Bally, Vlad²  Fernandez, Begona¹  Meda, Ana¹ 

[1] UNAM, Fac Ciencias, Dept Matemat, Coyoacan 04510, DF, Mexico

[2] Univ Paris Est Marne la Vallee, Lab Anal & Math Appl, UMR 8050, F-77454 Marne La Vallee 2, France

关键词: Ito processes;    Lower bounds for distributions;    Lower bounds for expectations;   

DOI  :  10.1016/j.spa.2011.05.004

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

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