Abstract
A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen’s method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen’s method and some well known second order methods and yields very promising results.
Similar content being viewed by others
References
Burrage, K., Burrage, P.M.: High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math. 22(1–3), 81–101 (1996). doi:10.1016/S0168-9274(96)00027-X. Special issue celebrating the centenary of Runge-Kutta methods
Burrage, K., Burrage, P.M.: Order conditions of stochastic Runge–Kutta methods by B-series. SIAM J. Numer. Anal. 38(5), 1626–1646 (2000). doi:10.1137/S0036142999363206 (electronic)
Burrage, K., Burrage, P.M., 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). doi:10.1098/rspa.2003.1247. Stochastic analysis with applications to mathematical finance
Burrage, P.M.: Runge–Kutta methods for stochastic differential equations. Ph.D. thesis, The University of Queensland, Brisbane (1999)
Debrabant, K., Kværnø, A.: B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181–203 (2008/2009). doi:10.1137/070704307
Debrabant, K., Kværnø, A.: Stochastic Taylor expansions: Weight functions of B-series expressed as multiple integrals. Stoch. Anal. Appl. 28(2), 293–302 (2010). doi:10.1080/07362990903546504
Debrabant, K., Rößler, A.: Continuous weak approximation for stochastic differential equations. J. Comput. Appl. Math. 214(1), 259–273 (2008). doi:10.1016/j.cam.2007.02.040
Debrabant, K., Rößler, A.: Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDEs and stability analysis. Appl. Numer. Math. 59(3–4), 595–607 (2009). doi:10.1016/j.apnum.2008.03.011
Debrabant, K., Rößler, A.: Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations. Appl. Numer. Math. 59(3–4), 582–594 (2009). doi:10.1016/j.apnum.2008.03.012
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, 2nd edn. Applications of Mathematics, vol. 21. Springer, Berlin (1999)
Komori, Y.: Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge-Kutta family. Appl. Numer. Math. 57(2), 147–165 (2007). doi:10.1016/j.apnum.2006.02.002
Komori, Y.: Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations. J. Comput. Appl. Math. 206(1), 158–173 (2007). doi:10.1016/j.cam.2006.06.006
Mackevičius, V., Navikas, J.: Second order weak Runge–Kutta type methods of Itô equations. Math. Comput. Simul. 57(1–2), 29–34 (2001). doi:10.1016/S0378-4754(00)00284-6
Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Mathematics and its Applications, vol. 313. Kluwer Academic, Dordrecht (1995). Translated and revised from the 1988 Russian original
Rößler, A.: Stochastic Taylor expansions for the expectation of functionals of diffusion processes. Stoch. Anal. Appl. 22(6), 1553–1576 (2004). doi:10.1081/SAP-200029495
Rößler, A.: Rooted tree analysis for order conditions of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations. Stoch. Anal. Appl. 24(1), 97–134 (2006). doi:10.1080/07362990500397699
Rößler, A.: Second order Runge–Kutta methods for Stratonovich stochastic differential equations. BIT 47(3), 657–680 (2007). doi:10.1007/s10543-007-0130-3
Rößler, A.: Second order Runge–Kutta methods for Itô stochastic differential equations. SIAM J. Numer. Anal. 47(3), 1713–1738 (2009). doi:10.1137/060673308 (electronic)
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 94–120 (1990). doi:10.1080/07362999008809220
Tocino, Á., Vigo-Aguiar, J.: Weak second order conditions for stochastic Runge–Kutta methods. SIAM J. Sci. Comput. 24(2), 507–523 (2002). doi:10.1137/S1064827501387814 (electronic)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anders Szepessy.
Rights and permissions
About this article
Cite this article
Debrabant, K. Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. Bit Numer Math 50, 541–558 (2010). https://doi.org/10.1007/s10543-010-0276-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-010-0276-2