On one-dimensional stochastic differential equations with unit diffusion coefficient. structure of solutions

The given paper deals with the construction and description of the integral funnel of all solutions of one-dimensional stochastic differential equations with unit diffusion. The notions of weak and strict regularity of the equations under consideration are proved to be equivalent and all the results known up to now on the existence and uniqueness of strict solutions follow from this. Specifically, it implies, that the weak solution with a distribution absolutely continuous with respect to the Wiener measure will automatically be strict. The solution of the innovation problem is shown to follow from this result in the case when the signal is a random variable. The essential part of the paper is devoted to the study of cases with non-unique solvability of equations, where instead of the condition of square integrability of the drift coefficient-a condition which is widely used in the theory of stochastic differential equations-we consider a Caratheodory condition (global and local) of the drift being integrable to the first power. This is classical in the theory of ordinary differential equations. Besides, the results on the integral funnel structure, which are known from the theory of deterministic equations, are applied to the stochastic case and this inevitably involves consideration of solutions with different type of measurability (strict, weak, anticipating). Specifically, minimal and maximal solutions are shown to be strict, while the solutions "filling up" the integral funnel are, generally speaking, anticipating. Still, one can show that the integral funnel can be described in terms of trajectories of strict solutions only, with each anticipating solution represented in the form of certain combinations of weak solutions. The main results are illustrated by a number of examples.