Abstract
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 (Debrabant and Rößler, Appl. Numer. Math. 59(3–4):595–607, 2009).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Notice that if R(p,q,ξ)=0 with a non-zero probability, then (3) is clearly numerically asymptotically stable.
In the implementation, we use the initializations \(K_{1}^{0}=X_{0}\) and \(K_{2}^{0}=X_{0}+(1-\gamma) h f(K_{1})\) and we consider the stopping criteria (\(\|K_{i}^{k+1}-K_{i}^{k}\|=0\) or \(\|K_{i}^{k+1}-K_{i}^{k}\|\geq\| K_{i}^{k}-K_{i}^{k-1}\|\)) which guaranties a convergence up to machine precision for the iterations (10). Other stopping criteria, such as \(\|K_{i}^{k+1}-K_{i}^{k}\|<\mathit{Tol}\) where Tol is a prescribed tolerance could also be considered.
References
Abdulle, A., Cirilli, S.: S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 30(2), 997–1014 (2008)
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)
Abdulle, A., Li, T.: S-ROCK methods for stiff Ito SDEs. Commun. Math. Sci. 6(4), 845–868 (2008)
Abdulle, A., Vilmart, G., Zygalakis, K.: Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput. (2013, to appear)
Alcock, J., Burrage, K.: Stable strong order 1.0 schemes for solving stochastic ordinary differential equations. BIT Numer. Math. 52(3), 539–557 (2012)
Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)
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)
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)
Gard, T.: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988)
Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Verlag Series in Comput. Math., vol. 8. Springer, Berlin (1993)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin (1996)
Higham, D.: A-stability and stochastic mean-square stability. BIT Numer. Math. 40, 404–409 (2000)
Higham, D.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000)
Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Komori, Y.: Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations. J. Comput. Appl. Math. 206(1), 158–173 (2007)
Komori, Y., Mitsui, T.: Stable ROW-type weak scheme for stochastic differential equations. Monte Carlo Methods Appl. 1(4), 279–300 (1995)
Milstein, G.N.: Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30(4), 750–766 (1986)
Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Scientific Computing. Springer, Berlin (2004)
Rößler, A.: Second order Runge-Kutta methods for Itô stochastic differential equations. SIAM J. Numer. Anal. 47(3), 1713–1738 (2009)
Saito, Y., Mitsui, T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 2254–2267 (1996)
Talay, D.: Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications. In: Lecture Notes in Control and Inform. Sci., vol. 61, pp. 294–313. Springer, Berlin (1984)
Tocino, A.: Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations. J. Comput. Appl. Math. 175(2), 355–367 (2005)
Acknowledgement
The research of A.A. is partially supported under Swiss National Foundation Grant 200021_140692.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anne Kværnø.
Rights and permissions
About this article
Cite this article
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. Bit Numer Math 53, 827–840 (2013). https://doi.org/10.1007/s10543-013-0430-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-013-0430-8