We consider the stochastic equation \[ X_t=x_0+\int _0^t b(u,X_{u})\mathrm{d}B_u,\quad t\ge 0, \] where $B$ is a one-dimensional Brownian motion, $x_0\in \mathbb{R}$ is the initial value, and $b\:[0,\infty )\times \mathbb{R}\rightarrow \mathbb{R}$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$, beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution.
We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.