## 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{eqnarray*} dX(t) &= & \left\{ a X(t)+ b X (qt) \right\} dt + \left\{ \sigma_1 + \sigma_2 X(t) + \sigma_3 X (q t) \right\}dW(t),\\ %[2mm] X(0) &= & X_0, \end{eqnarray*} 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 $\Theta$-Euler scheme ($\Theta \in [0,1]$). 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.

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### Key words

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

65C30, 65Q05.