Institut fur Stochastik, Friedrich-Schiller-Universitat, Ernst-Abbe Platz 1-4, D-07743 Jena, Germany, e-mail: engelbert@minet.uni-jena.de
Abstract: We consider the stochastic equation
X_t=x_0+\int_0^t b(u,X_u)\dd B_u,\quad t\geq0,
where $B$ is a one-dimensional Brownian motion, $x_0\in\Bbb R$ is the initial value, and $b [0,\infty)\times\Bbb R\to\Bbb R$ 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.
Keywords: one-dimensional stochastic equations, time-dependent diffusion coefficients, Brownian motion, existence of solutions, uniqueness in law, continuous local martingales, representation property
Classification (MSC 2000): 60H10, 60J60, 60J65, 60G44
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