A note on one-dimensional stochastic equations

Engelbert, Hans-Jürgen. "A note on one-dimensional stochastic equations." Czechoslovak Mathematical Journal 51.4 (2001): 701-712. <http://eudml.org/doc/30666>.

@article{Engelbert2001,
abstract = {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.},
author = {Engelbert, Hans-Jürgen},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {4},
pages = {701-712},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on one-dimensional stochastic equations},
url = {http://eudml.org/doc/30666},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Engelbert, Hans-Jürgen
TI - A note on one-dimensional stochastic equations
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 4
SP - 701
EP - 712
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/30666
ER -