Numerical Solution of Vasicek Equation by Using Brownian Wavelets and Multiple Ito-Integral

Document Type : بنیادی - نظری

Authors

1 Department of Mathematics, University of Ghom, Qom, 37161466711 Iran

2 Department of Mathematics, University of Qom, Qom, Iran

10.30473/coam.2021.54727.1146

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


‎‎bibitem{a6-4}‎
AndreMeyer P‎. ‎2000‎. ``‎Les processus stochastiques de 1950 a nos jours'', ‎In De Velopment of Mathematics 19502000‎, 813-848‎.
‎bibitem{a6-15}%15‎
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.‎
‎bibitem{a6-1}‎%1
Bachelier L‎. ‎1900‎. ``‎Theorie de la speculation''‎, ‎Gauthier-Villars‎, ‎Paris‎.‎‎‎‎‎
‎bibitem{a6-2}‎%2
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.‎
‎bibitem{a6-14}‎%14
Jorgensen P.E.T‎., ‎Song b M.S‎. ‎2007‎. ``‎Entropy encoding‎, ‎Hilbert space and Karhunen-Loeve transforms''‎, ‎Journal of Mathematical Physics 48‎.
‎ ‎bibitem{a6-5} ‎%5‎
Kahane J. P‎. ‎1997‎. ``‎A century of interplay between Taylor series''‎, ‎Fourier Series and Brownian Motion‎. ‎Bull‎. ‎London Math‎. ‎Soc.‎, ‎29(3), 257-279‎.‎‎
‎bibitem{a6-6}‎%6
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.
‎bibitem{a6-9}‎%9
‎‎Kendal B‎., ‎Sulem A‎. ``‎Applied stochastic control of jump diffusions'' (Universitext) 3r ed., 2019 Edition‎.
‎‎bibitem{a6-11}‎%11
Llie S‎. ‎2012‎. ``‎Variable time-stepping in the pathwise numerical solution of the chemical Langevin equation''‎, ‎Department of Mathematics‎, ‎Ryerson University‎, ‎Toronto‎, ‎Ontario‎, ‎M5B 2K3‎, ‎Canada THE Journal of Chemical Physics 137‎, ‎234110‎.
‎‎bibitem{a6-8} %8‎
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.
‎‎bibitem{a6-7}‎%7
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.
‎‎bibitem{a6-10}%10‎
Sarkka S‎. ‎2019. ``‎Applied stochastic differential equations''‎, ‎Aalto University‎, ‎Finland‎, ‎Cambridge University Press‎.
‎bibitem{a6-12}‎%12
Vercauteren N‎. ‎2006‎. ``‎Numerical investigation of solutions of Langevin equations''‎, ‎University of Berlin‎.‎‎
‎bibitem{a6-3}‎%3
Wiener N‎. ‎1923‎. ``‎Differential space'', Journal of Mathematics and Physics‎‎, ‎2, 132-174‎.
‎bibitem{a6-13}%13‎
Wu‎ ‎Y.‎, ‎Liang‎ ‎X‎. ‎2018. ``‎Vasicek model with mixed-exponential jumps and its applications in finance and insurance'', Advances in Difference Equations‎, ‎138‎.