On driftless one-dimensional sdes with time-dependent diffusion coefficients

Abstract

Consider a one-dimensional stochastic differential equation (S) without drift but with time-dependent diffusion coefficient. To obtain weak solutions, the time change method is applied: An increasing process is looked for such that a given Brownian motion, distorted by this process in its time argument, turns out to be a solution of (S). For this purpose, a time change equation(TC) has to be solved. We present one-to-one relations between (S) and (TC) concerning existence and uniqueness. Contrary to SDEs, pathwise uniqueness for (TC) coincides with that in law. By solving (TC), solutions of (S) are constructed under weak conditions admitting degenerate diffusion. The results improve those of Senf [18] and Rozkosz and Słomin ński [17]. Apart from degenerate diffusion, the main difference is that monotone approximation for the solutions of (TC), in contrast to weak convergence, is systematically exploited. As a consequence, the constructed solutions can be identified as pure so that they have the representation property