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October, 1991 Solutions of a Stochastic Differential Equation Forced Onto a Manifold by a Large Drift
G. S. Katzenberger
Ann. Probab. 19(4): 1587-1628 (October, 1991). DOI: 10.1214/aop/1176990225


We consider a sequence of $\mathbb{R}^d$-valued semimartingales $\{X_n\}$ satisfying $X_n(t) = X_n(0) + \int^t_0\sigma_n(X_n(s-))dZ_n(s) + \int^t_0F(X_n(s-))dA_n(s),$ where $\{Z_n\}$ is a "well-behaved" sequence of $\mathbb{R}^e$-valued semimartingales, $\sigma_n$ is a continuous $d \times e$ matrix-valued function, $F$ is a vector field whose deterministic flow has an asymptotically stable manifold of fixed points $\Gamma$, and $A_n$ is a nondecreasing process which asymptotically puts infinite mass on every interval. Many Markov processes with lower dimensional diffusion approximations can be written in this form. Intuitively, if $X_n(0)$ is close to $\Gamma$, the drift term $F dA_n$ forces $X_n$ to stay close to $\Gamma$, and any limiting process must actually stay on $\Gamma$. If $X_n(0)$ is only in the domain of attraction of $\Gamma$ under the flow of $F$, then the drift term immediately carries $X_n$ close to $\Gamma$ and forces $X_n$ to stay close to $\Gamma$. We make these ideas rigorous, give conditions under which $\{X_n\}$ is relatively compact in the Skorohod topology and give a stochastic integral equation for the limiting process(es).


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G. S. Katzenberger. "Solutions of a Stochastic Differential Equation Forced Onto a Manifold by a Large Drift." Ann. Probab. 19 (4) 1587 - 1628, October, 1991.


Published: October, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0749.60053
MathSciNet: MR1127717
Digital Object Identifier: 10.1214/aop/1176990225

Primary: 60H10
Secondary: 60J60 , 60J70

Keywords: diffusion , diffusion approximation , flow , Manifold , Semimartingale , Stochastic differential equation

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 4 • October, 1991
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