Geodesic
Continuous \(\Theta\)-methods for the stochastic pantograph equation
Electronic transactions on numerical analysis, Tome 11 (2000), pp. 131-151
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},
     year = {2000},
     volume = {11},
     zbl = {0968.65004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2000__11__a0/}
}
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TI  - Continuous \(\Theta\)-methods for the stochastic pantograph equation
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SP  - 131
EP  - 151
VL  - 11
UR  - http://geodesic.mathdoc.fr/item/ETNA_2000__11__a0/
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%A Buckwar,  Evelyn
%T Continuous \(\Theta\)-methods for the stochastic pantograph equation
%J Electronic transactions on numerical analysis
%D 2000
%P 131-151
%V 11
%U http://geodesic.mathdoc.fr/item/ETNA_2000__11__a0/
%G en
%F ETNA_2000__11__a0
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/