The Asymptotic Behavior of a Brownian Motion with a Drift from a Random Domain

Consider a Brownian motion with drift starting at an interior point of a random domain D in R ^d+1, d ≥ 1, let τ _D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of log P(τ _D  > T) are given for T → ∞, depending on the shape of the domain D and the order of the drift. The problem is motivated by the model in insurance and early works of Lifshits and Shi. The methods of proof are based on the calculus of variations and early works of Li, Lifshits and Shi in the drift free case.