Energy saving approximation of Wiener process under unilateral constraints

TL;DR

This paper investigates how to approximate Wiener processes efficiently under unilateral constraints, revealing that minimal energy consumption grows logarithmically with time and proposing an optimal adaptive strategy.

Contribution

It introduces a novel analysis of energy requirements for constrained Wiener process approximation and develops an adaptive strategy that achieves optimal energy efficiency.

Findings

Energy consumption grows logarithmically with time interval length.

An adaptive approximation strategy is constructed that is optimal within diffusion strategies.

The minimal energy needed for approximation is characterized almost surely over large intervals.

Abstract

We consider the energy saving approximation of a Wiener process under unilateral constraints. We show that, almost surely, on large time intervals the minimal energy necessary for the approximation logarithmically depends on the interval's length. We also construct an adaptive approximation strategy that is optimal in a class of diffusion strategies and also provides the logarithmic order of energy consumption.