## A note on order of convergence of numerical method for neutral stochastic functional differential equations.(English)Zbl 1257.65005

The authors consider $$n$$-dimensional neutral stochastic functional differential equations of the form $d[x(t)-u(x_{t})]=f(x_{t})dt+g(x_{t})dw(t),\;t\geq0,\;x(t)\in\mathbb{R}^{n},$ 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 $$\mathbb{C}([-\tau,0]).$$
The 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.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral 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)
Full Text:

### References:

 [1] Kolmanovskii, V.B.; Nosov, V.R., Stability and periodic modes of control systems with after effect, (1981), Moscow Nauka [2] Mao, X., Stochatic differential equations and applications, (1997), Horwood [3] Randielovic, J.; Jankovic, S., On the pth moment exponential stability criteria of neutral stochastic functional differential equations, J math anal appl, 326, 266-280, (2007) · Zbl 1115.60065 [4] Mao, X., Exponential stability in Mean square of neutral stochastic differential functional equations, Syst control lett, 26, 245-251, (1995) · Zbl 0877.93133 [5] Kolmanovskii, V.; Koroleva, N.; Maizenberg, T.; Mao, X.; Matasov, A., Neutral stochastic differential delay equation with Markovian switching, Stoch anal appl, 21, 839-867, (2003) · Zbl 1025.60028 [6] Kloeden, P.E.; Platen, E., Numerical solutions of stochastic differential equations, (1992), Springer Berlin · Zbl 0925.65261 [7] Küchler, U.; Pleten, E., Strong discrete time approximation of stochastic differential equations with time delay, Math comput simul, 54, 189-205, (2000) [8] Jiang, F.; Shen, Y.; Wu, F., Convergence of numerical approximation for jump models involving delay and Mean-reverting square root process, Stoch anal appl, 29, 216-236, (2011) · Zbl 1217.65012 [9] Jiang, F.; Shen, Y.; Liu, L., Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump, Commun nonlinear sci numer simulat, 16, 798-804, (2011) · Zbl 1221.60084 [10] Higham, D.J.; Mao, X.; Stuart, A.M., Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J numer anal, 40, 041-1063, (2002) [11] Yuan, C.; Mao, X., A note on the rate of convergence of the euler – maruyama method for scholastic differential equations, Stoch anal appl, 26, 325-333, (2008) · Zbl 1136.60040 [12] Jacob, N.; Wang, Y.; Yuan, C., Numerical solutions of stochastic differential delay equations with jumps, Stoch anal appl, 27, 825-853, (2009) · Zbl 1168.60356 [13] Mao, X., Numerical solutions of stochastic functional differential equations, LMSJ comput math, 6, 141-161, (2003) · Zbl 1055.65011 [14] Wu, F.; Mao, X., Numerical solutions of neutral stochastic functional differential equations, SIAM J numer anal, 46, 1821-1841, (2008) · Zbl 1173.65004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.