Continuous \(\Theta\)-methods for the stochastic pantograph equation. (English) Zbl 0968.65004

Baker, Christopher T. H.; Buckwar, Evelyn

ETNA, Electron. Trans. Numer. Anal. 11, 131-151 (2000).

The one-dimensional stochastic Itô pantograph equation \[ dX(t) = \{ a X(t) + b X(qt) \} dt + \{\sigma_1 +\sigma_2 X(t) +\sigma_3 X(qt)\} dW(t), \quad X(0) = X_0, \] for \(t\in [0, T]\) (\(T\) fixed and finite), driven by a Wiener process \(W\) and with deterministic delay parameter \(0<q<1\) is studied, both analytically and numerically. The authors discuss existence and uniqueness of strong solutions of this bilinear Itô stochastic delay differential equation. Finally, they study the mean square accuracy of approximations to the solution obtained by a continuous extension of the \(\Theta\)-Euler scheme with deterministic implicitness \(\Theta\in [0, 1]\). They prove the rate \(\gamma = 0.5\) of mean square convergence of these approximations using a bounded mesh of uniform step size \(h\), rising in the case of additive noise to \(\gamma = 1.0\). Some illustrative numerical experiments supplement this paper.

Reviewer: Henri Schurz (Minneapolis)

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness
34K50	Stochastic functional-differential equations

Keywords:

stochastic-numerical approximation; stochastic delay differential equations; stochastic pantograph equation; stochastic \(\Theta\)-methods; mean-square convergence rates; Itô equation; Euler scheme; numerical experiments