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Поредната сбирка на

Националния семинар по стохастика

ще се проведе на 17 октомври 2018 г. (сряда) от 14:00 в зала 403 на ИМИ-БАН с доклад на

Дончо Дончев (ФМИ – СУ)

на тема:

## Асимптотични решения на обратната задача на Ширяев

Резюме: 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.