Diffusion processes via parabolic equations: an infinitesimal approach to Lindeberg’s limit theorem

Heinz Weisshaupt

doi:10.4115/jla.2009.1.2

Abstract

We approach infinitesimal diffusion processes via a linkage to the diffu- sion equation. By this we obtain Lindeberg's limit theorem and a Lindeberg type limit theorem for diffusion processes by an application of the underspill principle. 2000 Mathematics Subject Classification 26E35 (primary); 60F05, 60J60, 35K99, 03H05 (secondary)

Highlights

The evolution of diffusions is in standard mathematics described either by ordinary stochastic differential equations, or by partial differential equations, called diffusion equations
Nonstandard diffusion theory is usually approached by linking the nonstandard stochastic process defined on a near interval by the concept of Loeb measure [25] to a corresponding standard stochastic process, as described in [8] or [35]
This approach to diffusion theory started with a nonstandard construction of the Brownian motion and the Ito integral in [1] and was further extended to stochastic integration in a broader context in [22] and applied to the analysis of the Ornstein-Uhlenbeck Process [23]

Summary

Introduction

The evolution of diffusions is in standard mathematics described either by ordinary stochastic differential equations, or by partial differential equations, called diffusion equations. Some further important developments concerning Loeb measures on Hyperfinite spaces can be found in [12] Their abstraction leads to the concept of neocompactness that can be used to prove the existence of solutions of stochastic differential equations with special properties [17]. Another way of linking nonstandard stochastic processes to the standard mathematical world - by Nelson’s reduction algorithm - is described in [27]. By this connection we further prove that the standard parts of the expectations of the infinitesimal diffusion processes are independent of the choice of the underlying infinitesimal model of white noise This fact can be interpreted as an infinitesimal version of a Lindeberg type limit theorem. (provided that the expectations E[f ◦ XT ], E[f ◦ YT ] and the solution u of (3) exist)

Preliminaries and Notation

Some probabilistic concepts

Diffusion Equations and Processes

Lindeberg’s Theorem and related results

An Example

A Appendix