Logarithmic Gradient Transformation and Chaos Expansion of Itô Processes

Abstract

Since the seminal work of Wiener (Am J Math 60:897–936, 1938), 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 on the initial condition. Therefore, the stochasticity of the transformed system lives in the initial condition and it can be treated conveniently with chaos expansion tools.

Keywords

Itô process / Chaos expansion / Fokker–Planck equation

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Funding

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung