×
Abdulle, A.; Vilmart, G.; Zygalakis, K. C.

Mean-square \(A\)-stable diagonally drift-implicit integrators of weak second order for stiff Itô stochastic differential equations. (English) Zbl 1367.65011

BIT 53, No. 4, 827-840 (2013).
Summary: We introduce two drift-diagonally-implicit and derivative-free integrators for stiff systems of Itô stochastic differential equations with general non-commutative noise which have weak order 2 and deterministic order 2, 3, respectively. The methods are shown to be mean-square \(A\)-stable for the usual complex scalar linear test problem with multiplicative noise and improve significantly the stability properties of the drift-diagonally-implicit methods previously introduced by K. Debrabant and A. Rößler [Appl. Numer. Math. 59, No. 3–4, 595–607 (2009; Zbl 1166.65304)].

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

Keywords:

drift-implicit stochastic methods; mean-square stability; stiff systems; Itô stochastic differential equations

Citations:

Zbl 1166.65304

Software:

S-ROCK; RODAS
Cite Review PDF
Full Text: DOI

References:

[1] Abdulle, A., Cirilli, S.: S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 30(2), 997-1014 (2008) · Zbl 1159.60329 · doi:10.1137/070679375
[2] Abdulle, A., Cohen, D., Vilmart, G., Zygalakis, K.C.: High order weak methods for stochastic differential equations based on modified equations. SIAM J. Sci. Comput. 34(3), 1800-1823 (2012) · Zbl 1246.65008 · doi:10.1137/110846609
[3] Abdulle, A., Li, T.: S-ROCK methods for stiff Ito SDEs. Commun. Math. Sci. 6(4), 845-868 (2008) · Zbl 1162.60330 · doi:10.4310/CMS.2008.v6.n4.a3
[4] Abdulle, A., Vilmart, G., Zygalakis, K.: Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput. (2013, to appear) · Zbl 1281.65005
[5] Alcock, J., Burrage, K.: Stable strong order 1.0 schemes for solving stochastic ordinary differential equations. BIT Numer. Math. 52(3), 539-557 (2012) · Zbl 1259.65002 · doi:10.1007/s10543-012-0372-6
[6] Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974) · Zbl 0278.60039
[7] Burrage, K., Burrage, P., Tian, T.: Numerical methods for strong solutions of stochastic differential equations: an overview. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460(2041), 373-402 (2004) · Zbl 1048.65004 · doi:10.1098/rspa.2003.1247
[8] Debrabant, K., Rößler, A.: Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDE: s and stability analysis. Appl. Numer. Math. 59(3-4), 595-607 (2009) · Zbl 1166.65304 · doi:10.1016/j.apnum.2008.03.011
[9] Gard, T.: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988) · Zbl 0628.60064
[10] Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Verlag Series in Comput. Math., vol. 8. Springer, Berlin (1993) · Zbl 0789.65048
[11] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin (1996) · Zbl 1192.65097 · doi:10.1007/978-3-642-05221-7
[12] Higham, D.: A-stability and stochastic mean-square stability. BIT Numer. Math. 40, 404-409 (2000) · Zbl 0961.65003 · doi:10.1023/A:1022355410570
[13] Higham, D.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753-769 (2000) · Zbl 0982.60051 · doi:10.1137/S003614299834736X
[14] Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992) · Zbl 0752.60043 · doi:10.1007/978-3-662-12616-5
[15] Komori, Y.: Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations. J. Comput. Appl. Math. 206(1), 158-173 (2007) · Zbl 1117.65010 · doi:10.1016/j.cam.2006.06.006
[16] Komori, Y., Mitsui, T.: Stable ROW-type weak scheme for stochastic differential equations. Monte Carlo Methods Appl. 1(4), 279-300 (1995) · Zbl 0841.65146 · doi:10.1515/mcma.1995.1.4.279
[17] Milstein, G.N.: Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30(4), 750-766 (1986) · Zbl 0596.60060 · doi:10.1137/1130095
[18] Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Scientific Computing. Springer, Berlin (2004) · Zbl 1085.60004 · doi:10.1007/978-3-662-10063-9
[19] Rößler, A.: Second order Runge-Kutta methods for Itô stochastic differential equations. SIAM J. Numer. Anal. 47(3), 1713-1738 (2009) · Zbl 1193.65006 · doi:10.1137/060673308
[20] 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
[21] Talay, D., Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications, No. 61, 294-313 (1984), Berlin · Zbl 0542.93077
[22] Tocino, A.: Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations. J. Comput. Appl. Math. 175(2), 355-367 (2005) · Zbl 1063.65009 · doi:10.1016/j.cam.2004.05.019
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.