Logarithmic Gradient Transformation and Chaos Expansion of Itô Processes

Since the seminal work of Wiener (Am J Math 60:897–936, 1938), chaos expansion has evolved to a powerful methodology for studying broad range stochastic differential equations. Yet its complexity systems subject white noise remains significant. The issue appears due fact that random increments generated by Brownian motion result in growing set variables with respect which process could be measured. In order cope this high dimensionality, we present novel transformation processes driven noise. particular, show under suitable assumptions, diffusion arising from can cast into logarithmic gradient induced measure process. Through transformation, resulting equation describes whose randomness depends only on initial condition. Therefore, stochasticity transformed system lives condition and it treated conveniently tools.