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Delay-dependent stability analysis of numerical methods for stochastic delay differential equations. (English) Zbl 1246.65013
The mean square asymptotic stability condition in terms of the coefficients of the linear stochastic delay differential equation (SDDE) has been given by J. A. D. Appleby, X. Mao and M. Riedle [Proc. Am. Math. Soc. 137, No. 1, 339–348 (2009; Zbl 1156.60045)], of which the deterministic part involves no delay. In the present paper, the mean square asymptotic stability of the numerical algorithm of a discretization approximation, so called $\theta$ method, of the above SDDE is studied, where the $\theta$ method uses the sum of weight $\theta$ and weight $1-\theta$ of the forward and backward discretization approximation respectively for the deterministic integral. The asymptotic stability condition is then obtained by analyzing the characteristic equation of the difference equation of the mean square sequence, It is proven that the backward Euler method preserves this property, while the Euler-Maruyama method preserves the instability property. Examples are illustrated.
##### MSC:
 65C30 Stochastic differential and integral equations 60H35 Computational methods for stochastic equations 60H10 Stochastic ordinary differential equations 34K50 Stochastic functional-differential equations 34K06 Linear functional-differential equations 65L20 Stability and convergence of numerical methods for ODE 65L12 Finite difference methods for ODE (numerical methods)
##### References:
 [1] Itô, K.; Nisio, M.: On stationary solutions of a stochastic differential equations, J. math. Kyoto univ. 4, 1-75 (1964) · Zbl 0131.16402 [2] Mohammed, S. -E.A.: Stochastic functional differential equations, (1984) [3] Mao, X.: Stochastic differential equations and applications, (1997) [4] Burrage, K.; Burrage, P.; Mitsui, T.: Numerical solutions of stochastic differential equations– implementation and stability issues, J. comput. Appl. math. 125, 171-182 (2000) · Zbl 0971.65003 · doi:10.1016/S0377-0427(00)00467-2 [5] Du, Q.; Zhang, T.: Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Numer. anal. 40, 1421-1445 (2002) · Zbl 1030.65002 · doi:10.1137/S0036142901387956 [6] Higham, D. J.: An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev. 43, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302 [7] Kloeden, P. E.; Platen, E.: Numerical solutions of stochastic differential equations, (1992) · Zbl 0752.60043 [8] Milstein, G. N.; Tretyakov, M.: Stochastic numerics for mathematical physics, (2004) [9] Platen, E.: An introduction to numerical methods for stochastic differential equations, Acta numer. 8, 197-246 (1999) · Zbl 0942.65004 [10] Schurz, H.: Numerical analysis of stochastic differential equations without tears, Handbook of stochastic analysis and applications, 237-359 (2002) · Zbl 0995.60052 [11] Zhao, W.; Tian, L.; Ju, L.: Convergence analysis of a splitting method for stochastic differential equations, Int. J. Numer. anal. Model. 5, 673-692 (2008) · Zbl 1163.65004 · doi:http://www.math.ualberta.ca/ijnam/Volume-5-2008/No-4-08/2008-04-08.pdf [12] Küchler, U.; Platen, E.: Strong discrete time approximation of stochastic differential equations with time delay, Math. comput. Simul. 54, 189-205 (2000) [13] 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 · doi:10.1112/S1461157000000322 · doi:http://www.lms.ac.uk/jcm/3/lms2000-002/ [14] 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 · doi:10.1016/j.cam.2005.01.018 [15] Liu, M.; Cao, W.; Fan, Z.: Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation, J. comput. Appl. math. 170, 255-268 (2004) · Zbl 1059.65006 · doi:10.1016/j.cam.2004.01.040 [16] Wu, F.; Mao, X.; Szpruch, L.: Almost sure exponential stability of numerical solutions for stochastic delay differential equation, Numer. math. 115, 681-697 (2010) · Zbl 1193.65009 · doi:10.1007/s00211-010-0294-7 [17] Bradul, N.; Shaikhet, L.: Stability of the positive point of equilibrium of Nicholson’s blowflies equation with stochastic perturbations: numerical analysis, Discrete dyn. Nat. soc. 2007 (2007) · Zbl 1179.60039 · doi:10.1155/2007/92959 [18] Shaikhet, L.; Roberts, J.: Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations, Adv. difference equ. 2006 (2006) · Zbl 1133.65003 · doi:10.1155/ADE/2006/73897 [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 · doi:10.1137/S0036142992228409 [20] Higham, D. J.: Mean-square and asymptotic stability of the stochastic theta methods, SIAM J. Numer. anal. 38, 753-769 (2000) · Zbl 0982.60051 · doi:10.1137/S003614299834736X [21] Guglielmi, N.: Delay dependent stability regions of ${\Theta }$-methods for delay differential equations, IMA J. Numer. anal. 18, 399-418 (1998) · Zbl 0909.65050 · doi:10.1093/imanum/18.3.399 [22] Guglielmi, N.; Hairer, E.: Order stars and stability for delay differential equations, Numer. math. 83, 371-383 (1999) · Zbl 0937.65079 · doi:10.1007/s002110050454 [23] Huang, C.: Delay-dependent stability of high order Runge–Kutta methods, Numer. math. 111, 377-387 (2009) · Zbl 1167.65045 · doi:10.1007/s00211-008-0197-z [24] 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 · doi:10.1016/j.cam.2005.11.035 [25] Buckwar, E.; Horvath-Bokor, R.; Winkler, R.: Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations, Bit 46, 261-282 (2006) · Zbl 1121.60071 · doi:10.1007/s10543-006-0060-5 [26] Appleby, J. A. D.; Mao, X.; Riedle, M.: Geometric Brownian motion with delay: mean square characterisation, Proc. amer. Math. soc. 137, 339-348 (2009) · Zbl 1156.60045 · doi:10.1090/S0002-9939-08-09490-2 [27] Lei, J.; Mackey, M. C.: Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM J. Appl. math. 67, 387-407 (2007) · Zbl 1167.34036 · doi:10.1137/060650234 [28] Baker, C. T. H.; Ford, N. J.: Some applications of the boundary-locus method and the method of D-partitions, IMA J. Numer. anal. 11, 143-158 (1991) · Zbl 0726.65152 · doi:10.1093/imanum/11.2.143 [29] Diekmann, O.; Van Gils, S. A.; Lunel, S. M. Verduin; Walther, H. -O.: Delay equations: functional-, complex-, and nonlinear analysis, (1995) [30] Huang, C.; Vandewalle, S.: An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays, SIAM J. Sci. comput. 25, 1608-1632 (2004) · Zbl 1064.65078 · doi:10.1137/S1064827502409717 [31] X. Qu, C. Huang, Delay-dependent exponential stability of the backward Euler method for nonlinear stochastic delay differential equations, Int. J. Comput. Math., in press (http://dx.doi.org/10.1080/00207160.2012.672731).