SUR LES SOLUTIONS FAIBLES D’EQUATIONS DIFFERENTIELLES
STOCHASTIQUES
Abstract: A characterization of the existence of the solutions of stochastics differential
equations on
- for any initial data - has been given by Engelbert and Schmidt in
[5]. In this paper we propose to give a necessary and sufficient condition for the
equation
 | (*) |
to admit a weak solution. By using a theorem of convergence for continuous local
martingals and its connection with the local time, we prove a lemma which generalizes
theorem (3) of [3]. Then we deduce that if equation (*) admits a solution verifying
then the diffusion coefficient
cannot vanish on strictly positive Lebesgue
measure set and, if
then
is locally integrable in a neighbourhood of
We finish with an extending the preceding results to equations with no zero
drift.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -