On the Weak Convergence of a Sequence of General Stochastic Difference Equations to a Diffusion,
Abstract
A convenient and useful method for showing weak convergence, to a diffusion, of the interpolated solutions of a (not necessarily Markovian) sequence of stochastic difference equations is developed. The technique involves the use of averaging methods to show that the weak limit satisfies the martingale problem of Strook and Varadhan which is associated with the diffusion. A truncation method is developed so that it is only necessary to work with the parts of the process before first escape from an arbitrary but bounded domain. The assumptions cover a wide variety of applications in systems theory, mathematical biology and elsewhere but the method of proof is adaptable to other special cases where our particular assumptions might not hold. Two applications are given in order to illustrate the relative ease of use of the method. The driving noise process in the difference equations can depend on the solution process of the difference equation, and one application where this is useful is given (a rate of convergence problem for simple stochastic approximations with sequentially averaged observations). (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1979
- Accession Number
- ADA078583
Entities
People
- Hai Huang
- Harold J. Kushner
Organizations
- Brown University