## Almost sure exponential stability of numerical solutions for stochastic delay differential equations.(English)Zbl 1193.65009

A theorem is proved that gives sufficient conditions for almost sure exponential stability (ASES) of Euler-Maruyama method numerical solutions of the $$n$$-dimensional nonlinear stochastic delay differential equation $dx(t)= f(x(t), x(t-\tau),t)\,dt+ g(x(t), x(t-\tau), t)\,dw(t),\quad t\geq 0.$ A counterexample is presented to show that without the linear growth condition on $$f$$ of the theorem, ASES may be lost. Then for the backward Euler-Maruyama method ASES is proved when a one-sided Lipschitz condition on $$f$$ in $$x$$ replaces the linear growth condition on $$f$$.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral equations 65C99 Probabilistic methods, stochastic differential equations

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### References:

 [1] 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 [2] Baker C.T.H., Buckwar E.: Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math. 184, 404–427 (2005) · Zbl 1081.65011 [3] Burrage K., Burrage P., Mitsui T.: Numerical solutions of stochastic differential equations–implematation and stability issues. J. Comput. Appl. Math. 125, 171–182 (2000) · Zbl 0971.65003 [4] Burrage K., Tian T.: A note on the stability propertis of the Euler methods for solving stochastic differential equations. N Z J. Math. 29, 115–127 (2000) · Zbl 0980.60083 [5] Hairer E., Wanner G.: Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996) · Zbl 0859.65067 [6] Higham D.J.: Mean-square and asymptotic stability of the stochastic theta methods. SIAM J. Numer. Anal. 38, 753–769 (2000) · Zbl 0982.60051 [7] Higham D.J., Mao X., Yuan C.: Almost sure and Moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45, 592–607 (2007) · Zbl 1144.65005 [8] Kloeden P.E., Platen E.: The Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992) · Zbl 0752.60043 [9] Liptser R.Sh., Shiryaev A.N.: Theory of Martingale. Kluwer Academic Publishers, Dordrecht (1989) · Zbl 0654.60035 [10] Mao X.: Approximate solutions for a class of stochastic evolution equations with variable delays–part II. Numer. Funct. Anal. Optim. 15, 65–76 (1994) · Zbl 0796.60068 [11] Mao X.: Exponential Stability of Stochastic Differential Equation. Marcel Dekker, New York (1994) · Zbl 0806.60044 [12] Mao X.: Stochastic Differential Equations and their Applications. Horwood, Chichester (1997) · Zbl 0892.60057 [13] Mao X.: Stochastic versions of the LaSalle theorem. J. Differ. Equ. 153, 175–195 (1999) · Zbl 0921.34057 [14] Mao X.: LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 236, 350–369 (1999) · Zbl 0958.60057 [15] Mao X.: The LaSalle-type theorems for stochastic differential equations. Nonlinear Stud. 7, 307–328 (2000) · Zbl 0993.60054 [16] Mao X.: A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268, 125–142 (2002) · Zbl 0996.60064 [17] Mao X.: Numerical solutions of stochastic functional differential equations. LMS J. Comput. Math. 6, 141–161 (2003) · Zbl 1055.65011 [18] Mao X.: Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math. 200, 297–316 (2007) · Zbl 1114.65005 [19] Mao X., Rassias M.J.: Khasminskii-type theorems for stochastic differential delay equations. Stoch. Anal. Appl. 23, 1045–1069 (2005) · Zbl 1082.60055 [20] Mao X., Yuan C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006) · Zbl 1126.60002 [21] Pang S., Deng F., Mao X.: Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math. 213, 127–141 (2008) · Zbl 1141.65006 [22] Rodkina A., Basin M.: On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term. Syst. Control Lett. 56, 423–430 (2007) · Zbl 1124.93066 [23] Rodkina A., Schurz H.: Almost sure asymptotic stability of drift-implicit {$$\theta$$}-methods for bilinear ordinary stochastic differential equations in $${\mathbb{R}^1}$$ . J. Comput. Appl. Math. 180, 13–31 (2005) · Zbl 1073.65009 [24] Saito Y., Mitsui T.: T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal. 2, 333–344 (1993) · Zbl 0834.65146 [25] Saito Y., Mitsui T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 2254–2267 (1996) · Zbl 0869.60052 [26] Shiryaev A.N.: Probability. Springer, Berlin (1996)
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