Logarithmic Gradient Transformation and Chaos Expansion of Ito Processes

TL;DR

This paper introduces a novel transformation for Ito processes driven by white noise, reducing their complexity by confining randomness to initial conditions, thus facilitating chaos expansion analysis.

Contribution

The authors propose a logarithmic gradient transformation that simplifies the analysis of white noise-driven stochastic processes by limiting stochasticity to initial conditions.

Findings

Transformation reduces dimensionality of stochastic systems

Chaos expansion becomes more tractable after transformation

Initial condition encapsulates all randomness in the transformed process

Abstract

Since the seminal work of Wiener, the chaos expansion has evolved to a powerful methodology for studying a broad range of stochastic differential equations. Yet its complexity for systems subject to the white noise remains significant. The issue appears due to the fact that the random increments generated by the Brownian motion, result in a growing set of random variables with respect to which the process could be measured. In order to cope with this high dimensionality, we present a novel transformation of stochastic processes driven by the white noise. In particular, we show that under suitable assumptions, the diffusion arising from white noise can be cast into a logarithmic gradient induced by the measure of the process. Through this transformation, the resulting equation describes a stochastic process whose randomness depends only upon the initial condition. Therefore the stochasticity of the transformed system lives in the initial condition and thereby it can be treated conveniently with the chaos expansion tools.