Quasi-invariant flow generated by Stratonovich SDE with BV drift coefficients

We generalize the results of Ambrosio [Invent. Math. 158 (2004), 227--260] on the existence, uniqueness and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. Then we construct an explicit solution to the corresponding stochastic transport equation in terms of the stochastic flow. The approximate differentiability of the flow is also studied when the drift is a Sobolev vector field.