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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 10, Fasc. 1,
pages 57 - 63
 

SUR LES SOLUTIONS FAIBLES D’EQUATIONS DIFFERENTIELLES STOCHASTIQUES

Ouknine Youssef

Abstract: A characterization of the existence of the solutions of stochastics differential equations on R - 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


           integral  t
Xt = x0 +   s(Xs)dBs
           0
(*)

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 <x> oo  = + oo , then the diffusion coefficient s cannot vanish on strictly positive Lebesgue measure set and, if s(x0) /= 0, then  -2
s  is locally integrable in a neighbourhood of x0. 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: -

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