Geodesic
Uniqueness of the inverse first-passage time problem and the shape of the Shiryaev boundary
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 717-744 Cet article a éte moissonné depuis la source Math-Net.Ru

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Given a distribution on the positive extended real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function such that the first passage time of this function by a reflected Brownian motion has the given distribution. We combine the ideas of Ekström and Janson, which were developed within the scope of the one-sided inverse first-passage time problem, with the methods of De Masi et al., which were used in the context of free boundary problems, in order to give a different proof for the uniqueness for the two-sided inverse first-passage time problem by using a stochastic order relation. We provide criteria for qualitative properties of solutions of the inverse first-passage problem, which apply to the boundary corresponding to the exponential distribution.
Keywords: inverse first-passage time, Brownian motion, Shiryaev problem, boundary crossing. 
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A. Klump; M. Kolb. Uniqueness of the inverse first-passage time problem and the shape of the Shiryaev boundary. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 717-744. http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a4/

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