The Annals of Probability

Solutions of a Stochastic Differential Equation Forced Onto a Manifold by a Large Drift

G. S. Katzenberger

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Abstract

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).

Article information

Source
Ann. Probab., Volume 19, Number 4 (1991), 1587-1628.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990225

Digital Object Identifier
doi:10.1214/aop/1176990225

Mathematical Reviews number (MathSciNet)
MR1127717

Zentralblatt MATH identifier
0749.60053

JSTOR
links.jstor.org

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Stochastic differential equation semimartingale diffusion diffusion approximation manifold flow

Citation

Katzenberger, G. S. Solutions of a Stochastic Differential Equation Forced Onto a Manifold by a Large Drift. Ann. Probab. 19 (1991), no. 4, 1587--1628. doi:10.1214/aop/1176990225. https://projecteuclid.org/euclid.aop/1176990225


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