# zbMATH — the first resource for mathematics

On the approximation of the solution of an anticipating stochastic differential equation. (Sur l’approximation de la solution d’une équation différentielle stochastique anticipative.) (French) Zbl 0949.60071
Let $$W$$ be the coordinate process on the classical Wiener space and $$G$$ an arbitrary random variable on this space. Given $$\alpha,b\in C^2_{l,b} (R)$$ with $$\sigma\sigma'\in C^2_{l,b}(R)$$ the authors prove for the anticipating stochastic differential equation with Stratonovich integral $X_t=G+\int^t_0 \sigma(X_s) \circ dW_t+\int^t_0b(X_s) ds,\quad t\in[0,1]$ (existence and uniqueness studied by D. Ocone and E. Pardoux in [Stochastic partial differential equations and applications. II., Lect. Notes Math. 1390, 197-204 (1989; Zbl 0703.60061)]) that, under some integrability assumption on $$\sigma,b$$ and their derivatives over $$R$$, the approximation by the solution of this equation with $$W$$ replaced by a linear interpolation converges to $$X$$ in Liouville spaces. The paper generalizes earlier results obtained by D. Feyel and A. de La Pradelle [Electron. J. Probab. 3, No. 7 (1998; Zbl 0901.60028)] in the case of a deterministic initial value.
Reviewer: R.Buckdahn (Brest)
##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H05 Stochastic integrals
Full Text:
##### References:
 [1] D. Feyel and A. de La Pradelle : ” On the approximate solutions of the Stratonovich equation ”. Electronic journal of Probability , vol. 3 , no. 7 , 1 - 14 ( 1998 ). MR 1624858 | Zbl 0901.60028 · Zbl 0901.60028 · emis:journals/EJP-ECP/EjpVol3/paper7.abs.html · eudml:119625 [2] D. Feyel and A. de La Pradelle : ” Fractional integrals and Brownian Processes ”. Prépublication 39 de l’université d’Evry Val d’Essone , ( 1996 ). [3] D. Feyel and A. de La Pradelle : ” On Fractional Brownian processes ”. ( 1999 ).To appear in Potential Analysis , MR 1696137 | Zbl 0944.60045 · Zbl 0944.60045 · doi:10.1023/A:1008630211913 [4] A. Millet , D. Nualart and M. Sanz-Solé : ” Large deviations for a class of anticipating stochastic differential equations ”. Annals of Probability , 20 , 1902 - 1931 ( 1992 ). Article | MR 1188048 | Zbl 0769.60053 · Zbl 0769.60053 · doi:10.1214/aop/1176989535 · minidml.mathdoc.fr [5] D. Ocone and E. Pardoux : ” A Generalized Itô-Ventzell formula. Applications to a class of anticipating stochastic differential equations ”. Annals Inst. H. Poincaré , 25 , 39 - 71 ( 1989 ). Numdam | MR 995291 | Zbl 0674.60057 · Zbl 0674.60057 · numdam:AIHPB_1989__25_1_39_0 · eudml:77339 [6] S.G. Samko , A.A. Kilbas and O.I. Marichev : ” Fractional integrals and Derivatives (Theory and Applications) ”. Gordon and Breach Science Publishers , ( 1987 ). MR 1347689 | Zbl 0818.26003 · Zbl 0818.26003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.