A note on one-dimensional stochastic equations

Engelbert, Hans-Jürgen

Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 701-712.

Résumé

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.

Détail

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