Stochastic differential equations for Dirichlet processes

We consider the stochastic differential equation dX₁ = a(X₁)dW₁ + b(X₁)dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and pathwise uniqueness results when a is in C^1/2 and b is only a generalized function, for example, the distributional derivative of a Hölder function or of a function of bounded variation. When b = aa¹, that is, when the generator of the SDE is the divergence form operator ℒ = 1/2 d/dx (a² d/dx), a result on non-existence of a strong solution and non-pathwise uniqueness is given as well as a result which characterizes when a solution is a semimartingale or not. We also consider extensions of the notion of Stratonovich integral.