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Keywords:
one-dimensional stochastic equations; time-dependent diffusion coefficients; Brownian motion; existence of solutions; uniqueness in law; continuous local martingales; representation property
one-dimensional stochastic equations; time-dependent diffusion coefficients; Brownian motion; existence of solutions; uniqueness in law; continuous local martingales; representation property
Summary:
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.
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