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The first exit time of a Brownian motion from an unbounded convex domain. (English) Zbl 1030.60032

Let \(\tau_D\) be the first exit time of a \((d+1)\)-dimensional Brownian motion from an unbounded open domain \(D= \{(x,y)\in \mathbb{R}^{d+1}\mid y> f(x),\;x\in \mathbb{R}^d\}\) starting at \((x_0,f(x_0)+ 1)\in \mathbb{R}^{d+1}\) for some \(x_0\in \mathbb{R}^d\), where the function \(f(.)\) on \(\mathbb{R}^d\) is convex and \(f(x)\to\infty\) as the Euclidean norm \(|x|\to\infty\). The author determines a general estimation for the asymptotics of \(\log P(\tau_D> t)\) by using Gaussian techniques.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60J65 Brownian motion
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