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The existing forward equation defines a probability current that disagrees with the random paths. A new one is adequate. With respect to that, the existing Ito paths consist of most probable increments (rather than mean ones), with Gaussian deviations. Further new features are the compliance with prediction theory and the agreement with physical steady states. Multiplicative noise, in particular the “spurious drift”, does no longer show up in the relevant equations. 

stochastic differential equations, multiplicative noise, Fokker-Planck equation, random paths, optimum prediction, steady states, transformation laws.