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The convergence of the Picard series for stochastic differential equations. (La convergence de la série de Picard pour les EDS (équations différentielles stochastiques).) (French) Zbl 0743.60056
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 343-354 (1989).
[For the entire collection see Zbl 0722.00030.]
The stochastic differential equation of the form \(X=a+FX\cdot Z\) is considered, where \(a\) is adapted cadlag, \(Z\) is a continuous semimartingale and \(F\) is a Lipschitz mapping from the class of adapted cadlag processes into itself. It is proved that if we define, for an arbitrary \(X_ 0\), the successive approximations as \(X_ n=a+FX_{n- 1}\cdot Z\), then they converge a.s., uniformly in \(t\) (globally!), their limit \(X\) being the unique solution of the equation. Moreover, almost surely, \[ \sup_ t| X_{n+1}(t)-X_ n(t)| \leq (k^ n/n!)^{1/2} \] for some \(k=k(\omega)\) and every \(n\geq n_ 0(\omega)\). An inequality of the same form holds for \(\sup| X_ n'(t)-X_ n(t)|,\) \((X_ n')_ n\) being the approximation sequence corresponding to another starting point \(X_ 0'\). These results are deduced from the fact that if \(Z=V+M\) is additionally assumed to satisfy \(| dV_ t|+d[M,M]_ t\leq dt\), then, for \(2\leq p < \infty\), \[ \|\sup_{s\leq t}| X_{n+1}(s)-X_ n(s)|\|_{L^ p} \leq Ce^{ct}((1+((2c)^{-1/2})^{2n}C_ p^{2n}K^{2n}(t^ n/n!))^{1/2}, \] provided that this inequality holds for \(n=0\), where \(K\) is the Lipschitz constant for \(F\) and \(C_ p\) is the constant from the inequality of Burkholder-Davis-Gundy. Again, the same is true for \(| X_ n'-X_ n|\).

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60E15 Inequalities; stochastic orderings
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