SOLUTIONS OF A STOCHASTIC DIFFERENTIAL-EQUATION FORCED ONTO A MANIFOLD BY A LARGE DRIFT

• 成果类型: Article
• 署名作者: KATZENBERGER, GS
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176990225
• 发表日期: 1991
• 页码: 1587-1628
• 关键词: 2 time scales; DIFFUSION APPROXIMATIONS; population-genetics; markov-chains; CONVERGENCE; SEQUENCES

摘要

We consider a sequence of R(d)-valued semimartingales {X(n)} satisfying X(n)(t) = X(n)(0) + integral-0/t-sigma-n(X(n)(s-))dZ(n)(s) + integral-0/t(F)(X(n)(s-))DA(n)(s), where {Z(n)} is a well-behaved sequence of R(e)-valued semimartingales, sigma-n is a continuous d x 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).