Given a Gaussian random walk X with drift, we consider the problem of estimating its first-passage time τ A for a given level A from an observation process Y correlated to X. Estimators may be any stopping times η with respect to the observation process Y. Two cases of the process Y are considered: a noisy version of X and a process X with delay d. For a given loss function f(x), in both cases we find exact asymptotics of the minimal possible risk E f((η − τ A )/r) as A, d → ∞, where r is a normalizing coefficient. The results are extended to the corresponding continuous-time setting where X and Y are Brownian motions with drift.
Problems of Information Transmission – Springer Journals
Published: Jul 13, 2012
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