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July, 1987 Asymptotic Properties of Some Multidimensional Diffusions
Charles R. Clark
Ann. Probab. 15(3): 985-1008 (July, 1987). DOI: 10.1214/aop/1176992076


Let $X_t \in \mathbf{R}^d$ be the solution to the stochastic differential equation $dX_t = \sigma(X_t) dB_t + b(X_t) dt, X_0 \in \mathbf{R}^d,$ where $B_t$ is a Brownian motion in $\mathbf{R}^d$. The aim of this paper is to make the following statement precise: "Let $x_t$ be a solution of $\dot{x} = b(x)$. If $|x_t| \rightarrow \infty$ as $t \rightarrow \infty$ and the drift vector field $b(x)$ is well behaved near $x_t$ then with positive probability, $X_t \rightarrow \infty$, and does so asymptotically like $x_t$." Examples are provided to illustrate the situations in which this theorem may be applied.


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Charles R. Clark. "Asymptotic Properties of Some Multidimensional Diffusions." Ann. Probab. 15 (3) 985 - 1008, July, 1987.


Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0622.60062
MathSciNet: MR893909
Digital Object Identifier: 10.1214/aop/1176992076

Primary: 60H10
Secondary: 60J25

Keywords: asymptotic behavior , diffusion process , Stochastic differential equation , transience

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 3 • July, 1987
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