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

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
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