On the relation between transformations of SDEs and of associated Kolmogorov equations

Abstract

The relationship between transformations of one-dimensional stochastic differential equations and those of the associated Kolmogorov backward equations is investigated. Stochastic transformations combining deterministic time changes, space-time diffeomorphisms, and Girsanov-type changes of probability measure are shown to be in correspondence with admissible transformations of the associated linear parabolic equations, namely the transformations preserving the parabolic structure up to the possible introduction of a zero-order potential term. A distinguished subclass of transformations, characterized by the absence of any potential term in the transformed Kolmogorov equation, is shown to coincide exactly with Doob transformations of the underlying probability measure of the SDE. This characterization further establishes a correspondence between Doob symmetries of an SDE and Lie point symmetries of the associated Kolmogorov backward equation. This correspondence, in turn, yields a constructive (probabilistic) Lie symmetry classification of autonomous Kolmogorov backward equations.