In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article

Authors

Department of Mathematics, University of Qom, Qom, Iran

Abstract

In this paper‎, ‎we present a new approach to solving stochastic differential equations and the Vasicek equation by using Brownian wavelets and multiple Ito-integral‎. ‎Firstly‎, ‎the calculation of the multiple Ito-integral based on the structure of Brownian motion is presented and the error of Ito-integrate computation is minimized under this condition‎. ‎Then‎, ‎the Brownian wavelets 1D and 3D based on coefficients Brownian motion are introduced‎. ‎After that‎, ‎a system of linear and nonlinear equations of coefficients Brownian motion is obtained such that by solving this system the approximate solution of the Vasicek equation is obtained‎. In the last section, ‎some numerical examples are given.

Keywords

[1] AndreMeyer P. 2000. “Les processus stochastiques de 1950 a nos jours”, In De Velopment of Mathematics 19502000, 813-848.
[2] Alouges F., de Bouard A., Merlet B. 2021. “Stochastic homogenization of the Landau–Lifshitz–Gilbert equation”, Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, ff10.1007/s40072-020-00185-4ff. ffhal-02020241f.
[3] Bachelier L. 1900. “Theorie de la speculation”, Gauthier-Villars, Paris.
[4] Einstein A. 1905. “On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat”, Annalen Der Physik, 17, 549-560.
[5] Jorgensen P.E.T., Song b M.S. 2007. “Entropy encoding, Hilbert space and Karhunen Loeve transforms”, Journal of Mathematical Physics 48.
[6] Kahane J. P. 1997. “A century of interplay between Taylor series”, Fourier Series and Brownian Motion. Bull. London Math. Soc., 29(3), 257-279.
[7] Kahane J.P. 1998. “Le mouvement brownien Un essai sur les origines de la theorie mathematique”, In Materiaux Pour Lhistoire Des Mathematiques au XX Siecle (Nice, 1996), Seminaires et Congres Volume: 3; 282.
[8] Kendal B., Sulem A. “Applied stochastic control of jump diffusions” (Universitext) 3r ed., 2019 Edition.
[9] Llie S. 2012. “Variable time-stepping in the path-wise numerical solution of the chemical Langevin equation”, Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada THE Journal of Chemical Physics 137, 234110.
[10] Mamon R.S. 2004. “Three Ways to Solve for Bond Prices in the Vasicek Model”, Department of Statistics, University of British Columbia Vancouver, BC, Canada V6T 1Z2, Journal of Applied Mathematics and Decision Sciences, 8(1), 114.
[11] Ozmen G., Ozsen S., Ylmaz B. 2016. “Denoising MR Images WithWeighted 3D- Discrete Wavelet Transform”, Electrical and Electronics Engineering, Faculty of Technology, 4th International Conference on Advanced Technology and Sciences (ICATRome) November 23-25.
[12] Sarkka S. 2019. “Applied stochastic differential equations”, Aalto University, Finland, Cambridge University Press.
[13] Vercauteren N. 2006. “Numerical investigation of solutions of Langevin equations”, University of Berlin.
[14] Wiener N. 1923. “Differential space”, Journal of Mathematics and Physics, 2, 132-174.
[15] Wu Y., Liang X. 2018. “Vasicek model with mixed-exponential jumps and its applications in finance and insurance”, Advances in Difference Equations, 138.