THE EXACT COUPLING WITH TRIVIAL COUPLING (COMBINED METHOD) IN TWO-DIMENSIONAL SDE WITH NON-INVERTIBLITY MATRIX

TitleTHE EXACT COUPLING WITH TRIVIAL COUPLING (COMBINED METHOD) IN TWO-DIMENSIONAL SDE WITH NON-INVERTIBLITY MATRIX
Publication TypeJournal Article
Year of Publication2019
AuthorsALNAFISAH YOUSEF
JournalDynamic Systems and Applications
Volume28
Issue1
Start Page111
Pagination32
Date Published01/2019
ISSN1056-2176
AMS Subject Classification60Hxx
Abstract

 In  Davie \cite{b2} paper, he assumed that the matrix $\big(b_{ik}(x)\big)$ is invertible for all $x$, but in this paper we will show how we could control the matrix which is non-invertible for some $x$ using the (\textit{Combined method}). We describe a method for non-invertibility case (\textit{Combined method}) and we investigate its convergence order which will give $O(h^{3/4}\sqrt{|\log(h)|})$ under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them.
PDFhttps://acadsol.eu/dsa/articles/28/1/7.pdf
DOI10.12732/dsa.v28i1.7
Refereed DesignationRefereed
Full Text

[1] A. Alfonsi, B. Jourdain and A. Kohatsu-Higa, Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme, Ann. Appl. Probab. 24 (2014), 1049-1080.
[2] A. Alfonsi, B. Jourdain and A. Kohatsu-Higa, Optimal transport bounds between the time-marginals of multidimensional diffusion and its Euler scheme, arXiv: 1405.7007.
[3] A. M. Davie, Pathwise approximation of stochastic differential equations using coupling, preprint: www.maths.ed.ac.uk/∼adavie/coum.pdf
[4] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag 1995.
[5] J. Koml´os, P. Major and G. Tusn´ady, An approximation of partial sums of independent RV’s and the sample DF. I, Z. Wahr. und Wer. Gebiete 32 (1975), 111-131.
[6] S. T. Rachev and L. Ruschendorff, Mass Transportation Problems, Volume 1, Theory; Volume 2, Applications. Springer-Verlag 1998.
[7] L. N. Vaserstein, Markov processes over denumerable products of spaces describing large system of automata (Russian), Problemy Peredaci Informacii 5 (1969), 64-72.
[8] M. Wiktorsson, Joint characteristic function and simultaneous simulation of iterated Itˆo integrals for multiple independent Brownian motions, Ann. Appl. Probab. 11 (2001), 470-487.
[9] T. Ryd´en and M.Wiktrosson, On the simulation of iteraled Ito integrals, Stochastic Processes Appl. 91 (2001), 151-168.
[10] S. Kanagawa, The rate of convergence for the approximate solutions of SDEs, Tokyo J. Math 12 (1989), 33-48.
[11] N. Fournier, Simulation and approximation of L´evy-driven SDEs, ESIAM Probab. Stat. 15 (2011), 233-248.
[12] E. Rio, Upper bounds for minimal distances in the central limit theorem, Ann. Inst. Henri Poincar´e Probab. Stat. 45 (2009), 802-817.
[13] A. B. Cruzeiro, P. Malliavin and A. Thalmaier, Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation, C. R. Math. Acad. Sci. Paris 338 (2004), 481-486.
[14] B. Charbonneau, Y. Svyrydov and P. Tupper,Weak convergence in the Prokhorov metric of methods for stochastic differential equations, IMA J. Numer. Anal. 30 (2010), 579-594.
[15] I. Gyongy and N. Krylov, Existence of strong solutions for Itˆo’s stochastic equations via approximations, Probab. Theory Related Fields 105 (1996), 143-158.
[16] Kloeden, P.E; Platen, E. andWright, I., The approximation of multiple stochastic integrals. J.Stoch. Anal. Appl. 10 (1992) 431-441.
[17] L.Hormander, Hypoelliptic second order differential equations, Acta. Math., 119 (1967), 147-171.
[18] Hairer, M. Malliavin’s proof of H¨ormander’s theorem, Bull Math. Sci., 135 (2011), no. 6-7, 650-666.
[19] E. Rio, Asymptotic constants for minimal distances in the central limit theorem, Elect. Comm. in Probab., 16 (2011), 96-103.
[20] A. Davie. KMT theory applied to approximations of SDE, In: Stochastic Analysis and Applications 2014, 185-201. Springer, 2014.
[21] C. Villani, Topics in Optimal Transportation, AMS, 2003.
[22] C. Villani, Optimal Transport Old and New, Springer-Verlag, 2009.