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Temps d’explosion d’équations différentielles stochastiques du type Doléans-Dade. (Explosion times for solutions of Doléans-Dade stochastic differential equations). (French) Zbl 0752.60048

For Doléans-Dade equations \(X_ t=x+\int^ t_ 0 A(X)_ s dZ_ s\) of the continuous martingale \(Z\), \(P(T(\omega,x)=\infty\), \(\forall x\in R^ d)=1\) is proved for the explosion time \(T(\infty,x)\) if the operator \(A\) is at most of linear growth and if for any stopping time \(T\), any \(u,v\in\{\omega:\sup_{[0,T]}|\omega_ t|\leq n\}\) for each \(n\), \[ \sup_{[0,T]}| A(u)-A(v)|\leq (\log n)^{1/2}\sup_{[0,T]}| u-v| \] holds.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
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