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

 Title THE EXACT COUPLING WITH TRIVIAL COUPLING (COMBINED METHOD) IN TWO-DIMENSIONAL SDE WITH NON-INVERTIBLITY MATRIX Publication Type Journal Article Year of Publication 2019 Authors ALNAFISAH YOUSEF Journal Dynamic Systems and Applications Volume 28 Issue 1 Start Page 111 Pagination 32 Date Published 01/2019 ISSN 1056-2176 AMS Subject Classification 60Hxx 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. PDF https://acadsol.eu/dsa/articles/28/1/7.pdf DOI 10.12732/dsa.v28i1.7 Refereed Designation Refereed 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.