The implementation of Milstein scheme in two-dimensional SDEs using the Fourier method. (English) Zbl 1470.65006

Alnafisah, Yousef

Abstr. Appl. Anal. 2018, Article ID 3805042, 7 p. (2018).

Summary: Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Software:

Matlab

References:

[1]	Kloeden, P. E.; Platen, E., Numerical solution of stochastic differential equations. Numerical solution of stochastic differential equations, Applications of Mathematics (New York), 23, (1992), Springer-Verlag, Berlin · Zbl 0752.60043 · doi:10.1007/978-3-662-12616-5
[2]	Rydén, T.; Wiktorsson, M., On the simulation of iterated Itô integrals, Stochastic Processes and Their Applications, 91, 1, 151-168, (2001) · Zbl 1047.60052 · doi:10.1016/S0304-4149(00)00053-3
[3]	Wiktorsson, M., Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions, The Annals of Applied Probability, 11, 2, 470-487, (2001) · Zbl 1019.60053 · doi:10.1214/aoap/1015345301
[4]	Davie, A. M., Pathwise approximation of stochastic differential equations using coupling
[5]	Kanagawa, S., The rate of convergence for approximate solutions of stochastic differential equations, Tokyo Journal of Mathematics, 12, 1, 33-48, (1989) · Zbl 0701.60056 · doi:10.3836/tjm/1270133546
[6]	Rachev, S. T.; Ruschendorff, L., Mass Transportation Problems, Volume 1, Theory; Volume 2, Applications, (1998), Springer-Verlag · Zbl 0990.60500
[7]	Komlós, J.; Major, P.; Tusnády, G., An approximation of partial sums of independent RV’s and the sample DF, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 32, 1-2, 111-131, (1975) · Zbl 0308.60029 · doi:10.1007/BF00533093
[8]	Fournier, N., Simulation and approximation of Lévy-driven stochastic differential equations, ESAIM: Probability and Statistics, 15, 233-248, (2011) · Zbl 1273.60080 · doi:10.1051/ps/2009017
[9]	Rio, E., Upper bounds for minimal distances in the central limit theorem, Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 45, 3, 802-817, (2009) · Zbl 1175.60020 · doi:10.1214/08-AIHP187
[10]	Rio, E., Asymptotic constants for minimal distance in the central limit theorem, Electronic Communications in Probability, 16, 96-103, (2011) · Zbl 1225.60047 · doi:10.1214/ECP.v16-1609
[11]	Alfonsi, A.; Jourdain, B.; Kohatsu-Higa, A., Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme, The Annals of Applied Probability, 24, 3, 1049-1080, (2014) · Zbl 1296.65010 · doi:10.1214/13-AAP941
[12]	Alfonsi, A.; Jourdain, B.; Kohatsu-Higa, A., Optimal transport bounds between the time-marginals of a multidimensional diffusion and its euler scheme, Electronic Journal of Probability, 20, article no. A070, (2015) · Zbl 1544.65024 · doi:10.1214/EJP.v20-4195
[13]	Cruzeiro, A. B.; Malliavin, P.; Thalmaier, A., Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation, Comptes Rendus Mathematique. Academie des Sciences, Paris, 338, 6, 481-486, (2004) · Zbl 1046.65006 · doi:10.1016/j.crma.2004.01.007
[14]	Charbonneau, B.; Svyrydov, Y.; Tupper, P. F., Weak convergence in the Prokhorov metric of methods for stochastic differential equations, IMA Journal of Numerical Analysis (IMAJNA), 30, 2, 579-594, (2010) · Zbl 1201.65008 · doi:10.1093/imanum/drn067
[15]	Vasershtein, L. N., Markov processes over denumerable products of spaces describing large system of automata, Problems of Information Transmission, 5, 3, 64-72, (1969) · Zbl 0273.60054
[16]	Gyöngy, I.; Krylov, N., Existence of strong solutions for Itô’s stochastic equations via approximations, Probability Theory and Related Fields, 105, 2, 143-158, (1996) · Zbl 0847.60038 · doi:10.1007/BF01203833
[17]	Davie, A., KMT theory applied to approximations of SDE, Stochastic Analysis and Applications, 100, 185-201, (2014), Springer · Zbl 1388.60116 · doi:10.1007/978-3-319-11292-3_7
[18]	Kloeden, P. E.; Platen, E.; Wright, I., The approximation of multiple stochastic integrals, Stochastic Analysis and Applications, 10, 4, 431-441, (1992) · Zbl 0761.60048 · doi:10.1080/07362999208809281