Convergence of an Adaptive Approximation Scheme for the Wiener Process

TL;DR

This paper introduces an adaptive approximation scheme for Wiener processes, where discretization points are dynamically chosen based on the process's deviation, and analyzes its limiting distribution as the deviation threshold approaches zero.

Contribution

It proposes a novel adaptive discretization method for Wiener processes and characterizes its asymptotic distribution as the approximation accuracy improves.

Findings

Limiting distribution of the approximation error is triangularly distributed.

The scheme adapts discretization points based on process deviation.

Convergence in distribution as eta approaches zero.

Abstract

The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable.