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We consider a multidimensional SDE with a Gaussian noise and a drift vectorbeing a vector function of bounded variation. We prove the existence ofgeneralized derivative of the solution with respect to the initial conditionsand represent the derivative as a solution of a linear SDE with coefficientsdepending on the initial process. The representation obtained is a naturalgeneralization of the expression for the derivative in the smooth case. Thetheory of continuous additive functionals is used.

A representation for the derivative with respect to the initial data of the solution of an SDE with a non-regular drift and a Gaussian noise