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Numerical solutions of neutral stochastic functional differential equations. (English) Zbl 1173.65004

The authors consider the numerical solution of neutral stochastic functional differential equations of the form \[ d[x(t)-u(x_{t})]=f(x_{t})dt+g(x_{t})dw(t),\;t\geq0, \] where \(x_{t}=\{x(t+\vartheta):-\tau\leq\vartheta\leq0\}\in\mathbb{C} ([-\tau,0];\mathbb{R}^{n})\) and the neutral term \(u(x_{t})\) is a contractive mapping with \(u(0)=0\).
The key contribution is to establish the strong mean square convergence theory of the Euler-Maruyama approximate solution under the local Lipschitz condition and the linear growth condition, [see X. Mao, LMS J. Comput. Math. 6, 141–161, electronic only (2003; Zbl 1055.65011)] as well.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K40 Neutral functional-differential equations
34K50 Stochastic functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Citations:

Zbl 1055.65011
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