## 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

### Keywords:

Brownian motion; Gaussian techniques
Full Text:

### References:

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