Alcock, Jamie; Burrage, Kevin Stable strong order 1.0 schemes for solving stochastic ordinary differential equations. (English) Zbl 1259.65002 BIT 52, No. 3, 539-557 (2012). A class of stable numerical methods (SSO1), an extension of the balanced method and the Milstein method, for solving systems of stiff Itô stochastic differential equations of the form \[ dX(t)= f(X(t))\,dt+ \sum^d_{i=1} g_i(X(t))\,dW^i(t),\quad X(t_0)= X_0, \] is presented. A theorem is proved that gives conditions under which strong convergence of order 1 to the exact solution is achieved. The theorem is then extended to establish conditions under which a generalized balanced scheme has strong order of convergence \(p\). A procedure for choosing the parameters in the SSO1 method to minimize global error is given. An examination of asymptotic stability is performed that shows that the SSO1 method surpasses the Milstein method, semi-implicit Milstein methods, and the balanced method. Reviewer: Melvin D. Lax (Long Beach) Cited in 8 Documents MSC: 65C30 Numerical solutions to stochastic differential and integral equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 65L04 Numerical methods for stiff equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations Keywords:stability; error bounds; balanced method; Milstein method; stiff Itô stochastic differential equations; convergence Software:S-ROCK; RODAS PDFBibTeX XMLCite \textit{J. Alcock} and \textit{K. Burrage}, BIT 52, No. 3, 539--557 (2012; Zbl 1259.65002) Full Text: DOI Link References: [1] Abdulle, J., Cirilli, S.: S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 30, 997–1014 (2008) · Zbl 1159.60329 · doi:10.1137/070679375 [2] Alcock, J., Burrage, K.: A note on the Balanced method. BIT Numer. Math. 46, 689–710 (2006) · Zbl 1116.65004 · doi:10.1007/s10543-006-0098-4 [3] Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974) · Zbl 0278.60039 [4] Burrage, K., Tian, T.: Implicit stochastic Runge-Kutta methods for stochastic differential equations. BIT Numer. Math. 44, 21–39 (2003) · Zbl 1048.65005 · doi:10.1023/B:BITN.0000025089.50729.0f [5] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996) · Zbl 0859.65067 [6] Higham, D.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38, 753–769 (2000) · Zbl 0982.60051 · doi:10.1137/S003614299834736X [7] Kahl, C., Schurz, H.: Balanced Milstein methods for ordinary SDEs. Monte Carlo Methods Appl. 12, 143–170 (2006) · Zbl 1105.65009 · doi:10.1515/156939606777488842 [8] Kloeden, P., Platen, E.: Higher order implicit strong numerical schemes for stochastic differential equations. J. Stat. Phys. 66, 283–314 (1992) · Zbl 0925.65261 · doi:10.1007/BF01060070 [9] Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations, 3rd edn. Springer, Berlin (2000) · Zbl 0752.60043 [10] Milstein, G.: Numerical Integration of Stochastic Differential Equations. Kluwer Academic, Norwell (1995) · Zbl 0810.65144 [11] Milstein, G., Platen, E., Schurz, H.: Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal. 35, 1010–1019 (1998) · Zbl 0914.65143 · doi:10.1137/S0036142994273525 [12] Sagirow, P.: Stochastic Methods in the Dynamics of Satellites. CISM Lecture Notes, vol. 57 (1970). Udine · Zbl 0335.70029 [13] 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 [14] Saito, Y., Mitsui, T.: Stability analysis of numeric schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 2254–2267 (1996) · Zbl 0869.60052 · doi:10.1137/S0036142992228409 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.