Geodesic
Continuous $\Theta$-methods for the stochastic pantograph equation
Electronic transactions on numerical analysis, Tome 11 (2000), pp. 131-151.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We consider a stochastic version of the pantograph equation: $dX(t) = {aX(t) + bX(qt)} dt + {\sigma 1 + \sigma 2X(t) + \sigma 3X(qt)}$ dW (t), $X(0) = X0$, for t $\in $[0, T ], a given Wiener process W and 0 q 1. This is an example of an It $\hat o$ stochastic delay differential equation with unbounded memory. We give the necessary analytical theory for existence and uniqueness of a strong solution of the above equation, and of strong approximations to the solution obtained by a continuous extension of $\surd $the $\Theta $-Euler scheme ($\Theta \in $[0, 1]). We establish $O( h)$ mean-square convergence of approximations obtained using a bounded mesh of uniform step h, rising in the case of additive noise to $O(h)$. Illustrative numerical examples are provided.
Classification : 65C30, 65Q05
Keywords: stochastic delay differential equation, continuous $\Theta $-method, mean-square convergence 
@article{ETNA_2000__11__a0,
     author = {Baker, Christopher T.H. and Buckwar, Evelyn},
     title = {Continuous $\Theta$-methods for the stochastic pantograph equation},
     journal = {Electronic transactions on numerical analysis},
     pages = {131--151},
     publisher = {mathdoc},
     volume = {11},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2000__11__a0/}
}
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Baker, Christopher T.H.; Buckwar, Evelyn. Continuous $\Theta$-methods for the stochastic pantograph equation. Electronic transactions on numerical analysis, Tome 11 (2000), pp. 131-151. http://geodesic.mathdoc.fr/item/ETNA_2000__11__a0/