Logarithmic Gradient Transformation and Chaos Expansion of Ito Processes
This provides a novel theoretical tool for mathematicians and physicists studying stochastic differential equations, potentially simplifying analysis of white-noise-driven systems.
The paper introduces a logarithmic gradient transformation that converts Ito processes driven by white noise into systems where randomness depends only on the initial condition, enabling efficient chaos expansion. This reduces the complexity of handling high-dimensional random variables in stochastic differential equations.
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.