| Title:
|
A note on one-dimensional stochastic equations (English) |
| Author:
|
Engelbert, Hans-Jürgen |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
701-712 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
one-dimensional stochastic equations |
| Keyword:
|
time-dependent diffusion coefficients |
| Keyword:
|
Brownian motion |
| Keyword:
|
existence of solutions |
| Keyword:
|
uniqueness in law |
| Keyword:
|
continuous local martingales |
| Keyword:
|
representation property |
| MSC:
|
60G44 |
| MSC:
|
60H10 |
| MSC:
|
60J60 |
| MSC:
|
60J65 |
| idZBL:
|
Zbl 1001.60059 |
| idMR:
|
MR1864037 |
| . |
| Date available:
|
2009-09-24T10:46:37Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127681 |
| . |
| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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