Research ArticleOpen AccessGoogle Scholar indexed
Approximation of an Integral Markov Process Arising in the Approximation of Stochastic Differential Equation
Department of Mathematics & Statistics, University of North Florida, Jacksonville, FL-32224, USA
- 1 Department of Mathematics & Statistics, University of North Florida, Jacksonville, FL-32224, USA
Advances in Pure Mathematics·Volume 12 (2022)·Pages 29–47·Published 25 January 2022·DOI10.4236/apm.2022.121003
Copy link · social · email
Abstract
We provide the derivation of a new formula for the approximation of an integral Markov process arising in the approximation of stochastic differential equations. This formula extends an existing formula derived in [1] . We have shown numerically that the leading order approximation of the differential equation with noise by solving an associated averaged problem and estimating the difference between them and the result is illustrated through some examples.
KeywordsVarianceMarkov ProcessParametric NoiseDifferential EquationApproximations
- Rahman, M. and Welfert, B. (2013) Functional Central Limit Theorem for Markov Processes and Chains. Journal of Probability and Statistical Science (JPSS), 11, 111-127.
- Battal Gazi Karakoc, S. and Zeybek, H. (2016) Solitary Wave Solutions of the GRLW Equation Using Septic B Spline Collocation Method. Applied Mathematics and Computation, 289, 159-172. https://doi.org/10.1016/j.amc.2016.05.021
- Zeybek, H. and Battal Gazi Karakoc, S. (2017) Application of the Collocation Method with B-Splines to the GEW Equation. Electronic Transactions on Numerical Analysis, 46, 77-88.
- Battal Gazi Karakoc, S., Geyikli, T. and Bashana, A. (2013) A Numerical Solution of the Modified Regularized Long Wave MRLW Equation Using Quartic B Splines. TWMS Journal of Applied and Engineering Mathematics, 3, 231-244. https://doi.org/10.1186/1687-2770-2013-27
- Turgut, A.K. and Battal Gazi Karakoc, S. (2018) A Numerical Technique Based on Collocation Method for Solving Modified Kawahara Equation. Journal of Ocean Engineering and Science, 3, 67-75. https://doi.org/10.1016/j.joes.2017.12.004
- Turgut, A.K., Battal Gazi Karakoc, S. and Triki, H. (2016) Numerical Simulation for Treatment of Dispersive Shallow Water Waves with Rosenau KdV Equation. The European Physical Journal Plus, 131, 1-15. https://doi.org/10.1140/epjp/i2016-16356-3
- Bhowmik, S. and Rahman, M. (2019) Stability and Accuracy Analysis of Theta Scheme for a Convolutional Integro-Differential Equation. Differential Equations and Dynamical Systems, 28, 633-646. https://doi.org/10.1007/s12591-019-00476-w
- Kloeden, P.E. and Platen, E. (1999) Numerical Solution of Stochastic Differential Equations, Applications of Mathematics. Vol. 23, Corrected Third Printing, Springer-Verlag, Berlin.
- Karlin, S. and Taylor, H.W. (1975) A First Course in Stochastic Processes. Academic Press, New York. https://doi.org/10.1016/B978-0-08-057041-9.50005-2
- Gikhmann (1973) Differential Equations with Random Functions. AMS Transl. 12.
- Borodin, A.N. and Salminen, P. (2002) Handbook Brownian Motion-Facts and Formulae. 2nd Edition, Probability and Its Applications, Birkhäuser. https://doi.org/10.1007/978-3-0348-8163-0
- Rahman, M. (2018) Asymptotic Estimate of Variance with Applications to Stochastic Differential Equations Arises in Mathematical Neuroscience. Communications in Statistics—Theory and Methods, 27, 289-306. https://doi.org/10.1080/03610926.2017.1303729