A note on order of convergence of numerical method for neutral stochastic functional differential equations (Q430392)

The authors consider \(n\)-dimensional neutral stochastic functional differential equations of the form NEWLINE\[NEWLINE d[x(t)-u(x_{t})]=f(x_{t})dt+g(x_{t})dw(t),\;t\geq0,\;x(t)\in\mathbb{R}^{n}, NEWLINE\]NEWLINE with initial data \(x_{0},\;x_{t}=\{x(t+\theta):-\tau\leq\theta\leq 0\}\in\mathbb{C}([-\tau,0])\), \(\;w(t)\) is an \(m\)-dimensional Brownian motion, \(f,\;g,\) and \(u\) are given functionals of the corresponding dimensions on \(\mathbb{C}([-\tau,0]).\)NEWLINENEWLINEThe authors study the order of convergence of the Euler-Maruyama method for such equations. They prove some convergence theorems both under the global and under the local Lipschitz conditions.