Abstract
A strongly converging method of order 2.5 for Ito stochastic differential equations with multidimensional nonadditive noise based on the unified Taylor–Stratonovich expansion is proposed. The focus is on the approaches and methods of mean square approximation of iterated Stratonovich stochastic integrals of multiplicities 1–5 the numerical simulation of which is the main difficulty in the implementation of the proposed numerical method.
Similar content being viewed by others
REFERENCES
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992).
G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics (Springer, Berlin, 2004).
M. Arato, Linear Stochastic Systems with Constant Coefficients. A Statistical Approach (Springer, Berlin, 1982).
A. N. Shiryaev, Foundations of Financial Mathematics (Fazis, Moscow, 1998), Vols. 1, 2 [in Russian].
R. Sh. Lipzer and A. N. Shiryaev, Statistics of Stochastic Processes: Nonlinear Filtering and Related Issues (Nauka, Moscow, 1974) [in Russian].
G. N. Milstein, Numerical Integration of Stochastic Differential Equations (Ural’skii Gos. Univ., Sverdlovsk, 1988) [in Russian].
P. E. Kloeden, E. Platen, and H. Schurz, Numerical Solution of SDE through Computer Experiments (Springer, Berlin, 1994).
D. F. Kuznetsov, Numerical Integration of Stochastic Differential Equations. 2 (Politekhnicheskii Univ., St. Petersburg, 2006) [in Russian]. https://doi.org/10.18720/SPBPU/2/s17-227
P. E. Kloeden and E. Platen, “The Stratonovich and Ito–Taylor expansions,” Math. Nachr. 151, 33–50 (1991).
D. F. Kuznetsov, “New representations of the Taylor–Stratonovich expansion,” Zap. Nauchn. Sem. POMI RAN. Veroyatn. Statist. 4. 278, 141–158 (2001).
O. Yu. Kul’chitskii and D. F. Kuznetsov, “Unified Taylor–Ito expansion,” Zap. Nauchn. Sem. POMI RAN. Veroyatn. Statist. 2. 244, 186–204 (1997).
D. F. Kuznetsov, “Development and application of the Fourier method for the numerical solution of Ito stochastic differential equations,” Comput. Math. Math. Phys. 58, 1058–1070 (2018).
D. F. Kuznetsov, Strong Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach (Politekhnicheskii Univ., St. Petersburg, 2011). https://doi.org/10.18720/SPBPU/2/s17-233
D. F. Kuznetsov, Stochastic Differential Equations: Theory and Practice of Numerical Solutions. With MATLAB Programs, 6th ed. Elektr. Zh. Differ. Uravn. Proc. Upravl., No. 4 (2018). https://diffjournal.spbu.ru/EN/numbers/2018.4/article.2.1.html
D. F. Kuznetsov, “Expansion of multiple Stratonovich stochastic integrals of fifth multiplicity, based on generalized multiple Fourier series,” 2018. arXiv:1802.00643 [math.PR].
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Applications (Naukova Dumka, Kiev, 1982) [in Russian].
P. E. Kloeden, E. Platen, and I. W. Wright, “The approximation of multiple stochastic integrals,” Stoch. Anal. Appl. 10, 431–441 (1992).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by A. Klimontovich
Rights and permissions
About this article
Cite this article
Kuznetsov, D.F. Explicit One-Step Numerical Method with the Strong Convergence Order of 2.5 for Ito Stochastic Differential Equations with a Multi-Dimensional Nonadditive Noise Based on the Taylor–Stratonovich Expansion. Comput. Math. and Math. Phys. 60, 379–389 (2020). https://doi.org/10.1134/S0965542520030100
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520030100