This paper examines the effect of varying stepsizes in finding the approximate solution of stochastic differential equations (SDEs). One step Milstein method (MLSTM) for solution of general first order stochastic differential equations (SDEs) has been derived using Itô Lemma and Euler-Maruyama Method as supporting tools. Two problems in the form of first order SDEs have been considered. The method of solution used is one step Milstein method. The absolute errors were calculated using the exact solution and numerical solution. Comparison of varying the stepsizes was achieved using mean absolute error criterion. The results showed that the mean absolute error due to approximation decreases as the stepsizes decreases. The order of convergence is approximately 1, which indicates the accuracy of the method. Also, the effect of varying stepsizes can also be identified using graphical method constructed for various stepsizes.
| Published in | Applied and Computational Mathematics (Volume 4, Issue 5) |
| DOI | 10.11648/j.acm.20150405.14 |
| Page(s) | 351-362 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2015. Published by Science Publishing Group |
Stochastic Differential Equations, Itô Lemma, Euler-Maruyama Method, Milstein Method, Wiener Process, Wiener Increment, Black Scholes Option Price Model, StepSizes
| [1] | Akinbo B.J., Faniran T. and Ayoola E.O. (2015). Numerical Solution of Stochastic Differential Equations. International Journal of Advanced Research in Science, Engineering and Technology. |
| [2] | Anna N. (2010). Economical Runge-Kutta Methods with week Second Order for Stochastic Differential Equations. Int. Contemp. Maths. Sciences, 5(24), 24 1151-1160. |
| [3] | Beretta, M., Carletti, F. and Solimano F. (2000). “On the Effects of Enviromental Flunctuationsin Simple Model of Bscteria- Bacteriophage Interaction, Canad. Appl. Maths. Quart. 8(4)321-366. |
| [4] | Bokor R.H. (2003). “Stochastically Stable One Step Approximations of Solutions of Stochastic Ordinary Differential Equations”, J. Applied Numerical Mathematics 44, 21-39. |
| [5] | Burrage, K. (2004). Numerical Methods for Strong Solutions of Stochastic Differential Equations: An overview. Proceedings: Mathematical Physical and Engineering Science, Published by Royal Society, 460(2041), 373-402. |
| [6] | Burrage, K. Burrage, P. and Mitsui T. (2000). Numerical Solutions of Stochastic Differential Equations-Implementation and Stability Issues. Journalof Computational&Applied Mathematics, 125, 171-182. |
| [7] | Fadugba S.E., Adegboyegun B.J., and Ogunbiyi O.T. (2013). On Convergence of Euler-Maruyama and Milstein Scheme for Solution of Stochastic Differential Equations. International Journal of Applied Mathematics and Modeling@ KINDI PUBLICATIONS.1(1), 9-15. ISSN: 2336-0054. |
| [8] | Higham D.J. (2001). An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. SIAM Review, 43(3), 525-546. |
| [9] | Lactus, M.L. (2008). Simulation and Inference for Stochastic Differential Equations withR Examples. Springer Science + Buisiness Media, LLC, 233 Springer Street, New York,NY10013, USA. Pp 61-62. |
| [10] | Oksendal B. (1998). Stochastic Differential Equations. An Introduction with Application, FifthEdition, Springer-Verlage, Berlin, Heidelberge, Italy. |
| [11] | Platen E. (1992). An Introduction to Numerical Methods of Stochastic Differential equations. ActaNumerica, 8, 197-246. |
| [12] | Rezaeyan R. and Farnoosh R. (2010). Stochastic Differential Equations and Application of Kalman-Bucy Filter in Modeling of RC Circuit. Applied Mathematical Sciences, Stochastic Differential Equations 4(33), 1119-1127. |
| [13] | Richardson M. (2009). Stochastic Differential Equations Case Study. (Unpublished).Sauer T. (2013). Computational Solution of Stochastic Differential Equations. WIRES ComputStat. doi: 10.1002/wics.1272. |
| [14] | Wang P. and Liu Z. (2009). “Stabilized Milstein Type Methods for Stiff Stochastic Systems”. Journal of Numerical Mathematics and Stochastics, Eulidean Press, LLC. 1(1): 33-34. |
| [15] | Yashihiro S. and Taketomo M. (1996). Stability Analysis of Numerical Schemes for Stochastic Differential Equations. SIAM J. Numer. Anal. 33(6), 2254. |
APA Style
Sunday Jacob Kayode, Akeem Adebayo Ganiyu. (2015). Effect of Varying StepSizes in Numerical Approximation of Stochastic Differential Equations Using One Step Milstein Method. Applied and Computational Mathematics, 4(5), 351-362. https://doi.org/10.11648/j.acm.20150405.14
ACS Style
Sunday Jacob Kayode; Akeem Adebayo Ganiyu. Effect of Varying StepSizes in Numerical Approximation of Stochastic Differential Equations Using One Step Milstein Method. Appl. Comput. Math. 2015, 4(5), 351-362. doi: 10.11648/j.acm.20150405.14
AMA Style
Sunday Jacob Kayode, Akeem Adebayo Ganiyu. Effect of Varying StepSizes in Numerical Approximation of Stochastic Differential Equations Using One Step Milstein Method. Appl Comput Math. 2015;4(5):351-362. doi: 10.11648/j.acm.20150405.14
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.acm.20150405.14,
author = {Sunday Jacob Kayode and Akeem Adebayo Ganiyu},
title = {Effect of Varying StepSizes in Numerical Approximation of Stochastic Differential Equations Using One Step Milstein Method},
journal = {Applied and Computational Mathematics},
volume = {4},
number = {5},
pages = {351-362},
doi = {10.11648/j.acm.20150405.14},
url = {https://doi.org/10.11648/j.acm.20150405.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150405.14},
abstract = {This paper examines the effect of varying stepsizes in finding the approximate solution of stochastic differential equations (SDEs). One step Milstein method (MLSTM) for solution of general first order stochastic differential equations (SDEs) has been derived using Itô Lemma and Euler-Maruyama Method as supporting tools. Two problems in the form of first order SDEs have been considered. The method of solution used is one step Milstein method. The absolute errors were calculated using the exact solution and numerical solution. Comparison of varying the stepsizes was achieved using mean absolute error criterion. The results showed that the mean absolute error due to approximation decreases as the stepsizes decreases. The order of convergence is approximately 1, which indicates the accuracy of the method. Also, the effect of varying stepsizes can also be identified using graphical method constructed for various stepsizes.},
year = {2015}
}
TY - JOUR T1 - Effect of Varying StepSizes in Numerical Approximation of Stochastic Differential Equations Using One Step Milstein Method AU - Sunday Jacob Kayode AU - Akeem Adebayo Ganiyu Y1 - 2015/09/09 PY - 2015 N1 - https://doi.org/10.11648/j.acm.20150405.14 DO - 10.11648/j.acm.20150405.14 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 351 EP - 362 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20150405.14 AB - This paper examines the effect of varying stepsizes in finding the approximate solution of stochastic differential equations (SDEs). One step Milstein method (MLSTM) for solution of general first order stochastic differential equations (SDEs) has been derived using Itô Lemma and Euler-Maruyama Method as supporting tools. Two problems in the form of first order SDEs have been considered. The method of solution used is one step Milstein method. The absolute errors were calculated using the exact solution and numerical solution. Comparison of varying the stepsizes was achieved using mean absolute error criterion. The results showed that the mean absolute error due to approximation decreases as the stepsizes decreases. The order of convergence is approximately 1, which indicates the accuracy of the method. Also, the effect of varying stepsizes can also be identified using graphical method constructed for various stepsizes. VL - 4 IS - 5 ER -
Email: