Volume 11, pp. 131-151, 2000.

Continuous Θ-methods for the stochastic pantograph equation

Christopher T. H. Baker and Evelyn Buckwar

Abstract

We consider a stochastic version of the pantograph equation: $$\begin{array}{rcl}dX(t)& =& \{aX(t)+bX(qt)\}dt+\{{\sigma }_1+{\sigma }_{2}X(t)+{\sigma }_{3}X(qt)\}dW(t),\\ X(0)& =& X_0,\end{array}$$ for $t\in [0,T]$, a given Wiener process $W$ and $0<q<1$. This is an example of an It\^{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 the $\mathrm{\Theta }$-Euler scheme ($\mathrm{\Theta }\in [0,1]$). We establish $\mathcal{O}(\sqrt{h})$ mean-square convergence of approximations obtained using a bounded mesh of uniform step $h$, rising in the case of additive noise to $\mathcal{O}(h)$. Illustrative numerical examples are provided.

Full Text (PDF) [602 KB], BibTeX

Key words

stochastic delay differential equation, continuous $\mathrm{\Theta }$-method, mean-square convergence.

AMS subject classifications

65C30, 65Q05.

ETNA articles which cite this article

Vol. 16 (2003), pp. 50-69 Henri Schurz: General theorems for numerical approximation of stochastic processes on the Hilbert space $H_2([0,T],\mu ,{\mathbf{R}}^d)$