Some Asymptotic Formulas for a Brownian Motion With a Regular Variation From a Parabolic Domain

Dawei Lu and Lixin Song

Communications in Statistics - Theory and Methods, 2015, vol. 44, issue 9, 1763-1778

Abstract: Consider a Brownian motion with a regular variation starting at an interior point of a domain D in Rd + 1, d ⩾ 1 and let τD denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of log P(τD > T) are given for T → ∞, depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the first exit time, and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.

Date: 2015
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DOI: 10.1080/03610926.2012.762397

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