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A strong quasi-invariance principle and a finite-dimensional integration byparts formula as in the Bismut approach to Malliavin calculus are obtainedthrough a suitable application of Lie's symmetry theory to autonomousstochastic differential equations. The main stochastic, geometrical andanalytical aspects of the theory are discussed and applications to someBrownian motion driven stochastic models are provided.

Integration by parts formulas and Lie's symmetries of SDEs