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Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. (English) Zbl 1059.65006
The authors prove conditions under which the semi-implicit Euler method has strong order of convergence 1/2 for scalar, linear stochastic differential delay equations. They also investigate mean square stability in terms of the stepsize of the method and the problem parameters. Numerical results illustrate some of the theory.

##### MSC:
 65C30 Stochastic differential and integral equations 60H10 Stochastic ordinary differential equations 34F05 ODE with randomness 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE 60H35 Computational methods for stochastic equations
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##### References:
 [1] Baker, C. T. H.; Buckwar, E.: Continuous ${\theta}$-methods for the stochastic pantograph equation. Electron. trans. Numer. anal. 11, 131-151 (2000) · Zbl 0968.65004 [2] Baker, C. T. H.; Buckwar, E.: Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. math. 3, 315-335 (2000) · Zbl 0974.65008 [3] Buckwar, E.: Introduction to the numerical analysis of stochastic delay differential equations. J. Cam 125, 297-307 (2000) · Zbl 0971.65004 [4] Burrage, K.; Burrage, P. M.: High strong order explicit Runge--Kutta methods for stochastic ordinary differential equations. Appl. numer. Math. 22, 81-101 (1996) · Zbl 0868.65101 [5] Higham, D. J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. anal. 38, 753-769 (2000) · Zbl 0982.60051 [6] Y. Hu, S.E.A. Mohammed, F. Yan, Discrete-time approximations of stochastic delay equations: the Milstein Scheme, Ann. Probab. 32 (2004) 265--314. · Zbl 1062.60065 [7] Kloeden, P. E.; Platen, E.: Numerical solution of stochastic differential equations. (1992) · Zbl 0752.60043 [8] Kolmanovskii, V.; Myshkis, A.: Applied theory of fundamental differential equations. (1992) · Zbl 0917.34001 [9] Kolmanovskii, V.; Shaikhet, L.: General method of Lyapunov functionals construction for stability investigation of stochastic difference equations, in: dynamical systems and applications. (1995) [10] Küchler, U.; Platen, E.: Strong discrete time approximation of stochastic differential equations with time delay. Math. comput. Simulation 54, 189-205 (2000) [11] Küchler, U.; Platen, E.: Weak discrete time approximation of stochastic differential equations with time delay. Math. comput. Simulation 59, 497-507 (2002) · Zbl 1001.65005 [12] Mao, X.: Exponential stability of stochastic differential equations. (1994) · Zbl 0806.60044 [13] Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stochastic process. Appl. 65, 233-250 (1996) · Zbl 0889.60062 [14] Mao, X.: Stochastic differential equations and applications. (1997) · Zbl 0892.60057 [15] Mao, X.; Sabanis, S.: Numerical solutions of stochastic differential delay equations under local Lipschitz condition. J. Cam 151, 215-227 (2003) · Zbl 1015.65002 [16] Milstein, G. N.: Numerical integration of stochastic differential equations. (1995) · Zbl 0810.65144 [17] S.E.A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, Vol. 99, Pitman, London, 1984. · Zbl 0584.60066 [18] Saito, Y.; Mitsui, T.: T-stability of numerical schemes for stochastic differential equations. World sci. Ser. appl. Anal. 2, 333-344 (1993) · Zbl 0834.65146 [19] Saito, Y.; Mitsui, T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. anal. 33, 2254-2267 (1996) · Zbl 0869.60052 [20] Tian, T. H.; Burrage, K.: Two-stage stochastic Runge--Kutta methods for stochastic differential equations. BIT numer math. 42, 625-643 (2002) · Zbl 1016.65002