In this work, a one-step method of Euler-Maruyama (EMM) type has been
developed for the solution of general first order stochastic differential
equations (SDEs) using Ito integral equation as basis tool. The effect of
varying stepsizes on the numerical solution is also examined for the SDEs. Two
problems of first order SDEs are solved. Absolute errors for the problems are
obtained from which the mean absolute errors (MAEs) are calculated. Comparison
of variation in stepsizes is achieved using the MAEs. The results show that the
MAEs decrease as the stepsize decreases. The strong orders of convergence and
the residuals for the problem for the theoretical are respectively obtained
using Least Square Fit. This work produces numerical values for the solution to
the problems which differ from the existing methods of EMM type in which results
are always obtained by simulation.
Cite this paper
Kayode, S. J. , Ganiyu, A. A. and Ajiboye, A. S. (2016). On One-Step Method of Euler-Maruyama Type for Solution of Stochastic Differential Equations Using Varying Stepsizes. Open Access Library Journal, 3, e2247. doi: http://dx.doi.org/10.4236/oalib.1102247.
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,
1119-1127.
Awoyemi, D.O. (1999) A Class of Continuous Linear Methods
for General Second Order Initial Value Problems In Ordinary Differential Equations. International Journal of Computer
Mathematics, 72, 29-37. http://dx.doi.org/10.1080/00207169908804832
Awoyemi, D.O. (2001) A New
Sixth-Order Algorithms for General Second Order Ordinary Differential Equations. International Journal of Computer
Mathematics, 77, 177-124. http://dx.doi.org/10.1080/00207160108805054
Awoyemi, D.O. (2003) A
P-Stable Linear Multistep Method for Solving Third Order Ordinary Differential
Equations. International Journal of
Computer Mathematics, 80, 85-991. http://dx.doi.org/10.1080/0020716031000079572
Awoyemi, D.O. (2005) An
Algorithm Collocation Methods for Direct Solution of Special and General Fouth
Order Initial Value Problems of Ordinary Differential Equations. Journal of the Nigerian Association of Mathematical
Physics, 6, 271-238.
Awoyemi, D.O., Kayode, S.J. and Adoghe, L.O. (2013) A Four-Point Fully
Implicit Method for the Numerical Integration of Third-Order Ordinary Differential Equations. InternationalJournal of
Physical Sciences, 9, 7-12. http://dx.doi.org/10.5897/IJPS2013.4019
Kayode, S.J. (2005) An Improved Numerov Method for
Direct Solution of General Second Order Initial Value Problems of Ordinary
Differential Equations. Proc. Semiar, Nat. Math. Centre, Abuja.
Kayode, S.J. (2009) A Zero Stable Method
for Direct Solution of Fourth Order Ordinary Differential Equations. American Journal of Applied Sciences, 5,
1461-1466.
Kayode, S.J. (2011) A Class of One-Point Zero-Stable Continuous
Hybrid Methods for Direct Solution of Second Order Differential Equations. African Journal of Mathathematicsand Computer Science Researche, 4, 93-99.
Kayode, S.J. and Adeyeye, O. (2011) A 3-Step Hybrid Method for
Direct Solution of Second Order Initial Value Problems. Australian Journal of basic and Applied Sciences, 5, 2121-2126.
Kayode, S.J. and Adeyeye, O. (2013) Two-Step Two-Point
Hybrid Methods for General Second Order Differential Equations. African Journal of Mathematics and Computer
Science Research, 6, 191-196.
Kayode, S.J. and Obarhua, F.O. (2013) Continuous
Y-Function Hybrid Methods for Direct Solution of Differential Equations. International Journal of Differential Equations andApplications, 12, 37-48.
Kayode, S.J. (2014) Symmetric Implicit
Multiderivative Numerical Integrators for Direct Solution of Fifth-Order Differential
Equations. Thammasat International
Journal of Science and Technology, 19, 1-8.
Kayode, S.J., Duromola, M.K. and Bolaji, B. (2014) Direct Solution of Initial
Value Problems of Fourth Order Ordinary Differential Equations Using Modified
Implicit Hybrid Block Method. Journal of
Scientific Research and Reports, 3, 2790-2798. http://dx.doi.org/10.9734/jsrr/2014/11953
Olabode, B.T. (2007) Some
Linear Multistep Methods for Special General Third Order Initial Value Problems
of Ordinary Differential Equations. PhD Thesis, Federal University of Technology,
Akure. (Unpublished)
Olabode, B.T. (2013) A Block Predictor-Corrector
Method for the Direct Solution of General
Fifth Order Ordinary Differential Equations. Canadian Journal on Computing inMathematics, Natural Science, Engineering and Medicine, 4, 133-139.
Olabode, B.T. (2013) Block Multistep
Method for Direct Solution of Third Order Ordinary Differential Equations. FUTA Journal of Research in Science, 2, 194-200.
Onumanyi, P., Sirisena,
U.W. and Dauda, Y. (2001) Towards Uniformly Accurate Continuous
Finite Difference Approximation of Ordinary Differential Equations. Bagale
Journal of Pure and Applied Sciences, 1, 5-8.
Yahaya, Y.A. and
Badmus, A.M. (2009) A Class of Collocation Methods for General Second
Order Ordinary Differential Equations. African
Journal of Mathematics and Computer Science Research, 2, 69-72.
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, 608-616.
Anna, N. (2010) Economical Runge-Kutta Methods with Week Second Order
for Stochastic Differential Equations. International
Journal of Contemporary Mathematical Sciences, 5, 1151-1160.
Burrage,
K. (2004) Numerical Methods for Strong Solutions of Stochastic Differential
Equations: An Overview. Proceedings of
theMathematical Physical and
Engineering Science, 460,
373-402.
Burrage, K., Burrage, P. and Mitsui, T. (2000)
Numerical Solutions of Stochastic Differential Equations-Implementation and Stability Issues. Journal of Computational &Applied Mathematics, 125, 171-182. http://dx.doi.org/10.1016/S0377-0427(00)00467-2
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,1, 9-15.
Kayode, S.J. and
Ganiyu, A.A. (2015) Effect of Varying Stepsize in Numerical Approximation
of Stochastic Differential Equations Using One Step Milstein Method. Applied and Computational Mathematics, 4, 351-362. http://dx.doi.org/10.11648/j.acm.20150405.14
Saito, Y. and
Mitsui, T. (1996) Stability Analysis of Numerical Schemes for Stochastic Differential
Equations. SIAM Journal on Numerical
Analysis, 33, 2254-2267. http://dx.doi.org/10.1137/s0036142992228409
Lactus,
M.L. (2008) Simulation and Inference for Stochastic Differential Equations with
R Examples. Springer Science Buisiness Media, LLC, New York, 61-62.