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The next meeting of

the National Seminar on Probability and Statistics

will be held on  October 17, 2018 (Wednesday) at 2 p.m. in Room 403 of IMI-BAS. A talk will be delivered by

Doncho Donchev (FMI, Sofia University)


Asymptotic Solutions of Shiryaev’s Inverse Problem

Abtract: In our recent paper, we characterized the exit density of a Brownian motion through one-sided smooth boundaries in terms of a solution of some parabolic second-order PDE. It turns out that this equation can be reduced to a first-order PDE. It is shown that the solution to the last equation admits an analytic representation only for three classes of boundaries – parabolic boundaries, square-root boundaries and rational boundaries. Our approach is substantiated by an example, where we find the exit density of a boundary not studied so far.

Next, we discuss the inverse first exit problem. We derive an asymptotic formula, that describes the small time behaviour of the exit density p_f(t) that corresponds to a boundary f(t). Making use of this formula, we construct a function f(t) such that log(p_f(t))=log(p_eta(t))+ o(t) for a large family of densities p_eta(t) of non-negative random variables.