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A note on one-dimensional stochastic equations
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 701-712 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 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.
Classification : 60G44, 60H10, 60J60, 60J65
Keywords: one-dimensional stochastic equations; time-dependent diffusion coefficients; Brownian motion; existence of solutions; uniqueness in law; continuous local martingales; representation property 
@article{CMJ_2001_51_4_a3,
     author = {Engelbert, Hans-J\"urgen},
     title = {A note on one-dimensional stochastic equations},
     journal = {Czechoslovak Mathematical Journal},
     pages = {701--712},
     year = {2001},
     volume = {51},
     number = {4},
     mrnumber = {1864037},
     zbl = {1001.60059},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a3/}
}
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Engelbert, Hans-Jürgen. A note on one-dimensional stochastic equations. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 701-712. http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a3/