Application of Ant Colony Programming Approach for Solving Systems of Stochastic Differential Equations

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DOI: https://doi.org/10.28924/2291-8639-22-2024-105
Published: Jun 24, 2024
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Ali Sami Rashid, Salah H Abid, Sadiq A. Mehdi, Application of Ant Colony Programming Approach for Solving Systems of Stochastic Differential Equations, Int. J. Anal. Appl., 22 (2024), 105.

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Ali Sami Rashid, Salah H Abid, Sadiq A. Mehdi

Abstract

Stochastic differential equations (SDE) have wide applications in natural phenomena, engineering, finance, and biological models. Obtaining analytic solutions for an SDE is often complex, and the complexity increases for an SDE system. The paper introduces ant colony programming (ACP) as a novel approach for solving SDE system. Ant colony programming was developed in two directions, the first is to add the Wiener process  as a variable to the terminals and functions, and the second is to construct the appropriate fitness function . ACP constructs mathematical expressions and evaluates them using the fitness function . The ACP proposed effectiveness has been demonstrated by applying to 2,3 and 4-dimensional SDE systems. The most important finding of this work is that ACP generates symbolic stochastic processes that represent solutions for SDE system. Methods for solving SDE systems are important tools for study phenomena that involve noise or randomness.

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References

  1. W.J. de A. Lobão, M.A.C. Pacheco, D.M. Dias, A.C.A. Abreu, Solving Stochastic Differential Equations Through Genetic Programming and Automatic Differentiation, Eng. Appl. Artif. Intell. 68 (2018), 110–120. https://doi.org/10.1016/j.engappai.2017.10.021.
  2. S. Lototsky, B. Rozovskii, Stochastic Differential Equations: A Wiener Chaos Approach, in: From Stochastic Calculus to Mathematical Finance, Springer Berlin Heidelberg, Berlin, Heidelberg, 2006: pp. 433–506. https://doi.org/10.1007/978-3-540-30788-4_23.
  3. C.C. Chen, K. Yao, Stochastic-Calculus-Based Numerical Evaluation and Performance Analysis of Chaotic Communication Systems, IEEE Trans. Circuits Syst. I 47 (2000), 1663–1672. https://doi.org/10.1109/81.899918.
  4. L. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, Rhode Island, 2013. https://doi.org/10.1090/mbk/082.
  5. S.D. Jacka, B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, J. Amer. Stat. Assoc. 82 (1987), 948. https://doi.org/10.2307/2288814.
  6. M. Bayram, T. Partal, G. Orucova Buyukoz, Numerical Methods for Simulation of Stochastic Differential Equations, Adv. Differ. Equ. 2018 (2018), 17. https://doi.org/10.1186/s13662-018-1466-5.
  7. A.F. Fareed, M.S. Semary, H.N. Hassan, Two Semi-Analytical Approaches to Approximate the Solution of Stochastic Ordinary Differential Equations with Two Enormous Engineering Applications, Alexandria Eng. J. 61 (2022), 11935–11945. https://doi.org/10.1016/j.aej.2022.05.054.
  8. X. Mao, The Truncated Euler–Maruyama Method for Stochastic Differential Equations, J. Comput. Appl. Math. 290 (2015), 370–384. https://doi.org/10.1016/j.cam.2015.06.002.
  9. Z. Yin, S. Gan, An Improved Milstein Method for Stiff Stochastic Differential Equations, Adv. Differ. Equ. 2015 (2015), 369. https://doi.org/10.1186/s13662-015-0699-9.
  10. P. Wang, Three-stage stochastic Runge-Kutta methods for stochastic differential equations, J. Comput. Appl. Math. 222 (2008), 324–332. http://doi.org/10.1016/j.cam.2007.11.001.
  11. D.F. Anderson, D.J. Higham, Y. Sun, Multilevel Monte Carlo for Stochastic Differential Equations with Small Noise, SIAM J. Numer. Anal. 54 (2016), 505–529. https://doi.org/10.1137/15M1024664.
  12. A.R. Soheili, F. Soleymani, Iterative Methods for Nonlinear Systems Associated with Finite Difference Approach in Stochastic Differential Equations, Numer. Algor. 71 (2016), 89–102. https://doi.org/10.1007/s11075-015-9986-5.
  13. M. Boryczka, Z.J. Czech, Solving Approximation Problems by Ant Colony Programming, in: Proceedings of the 4th Annual Conference on Genetic and Evolutionary Computation, 2002.
  14. M. Boryczka and W. Wiezorek, Solving Approximation Problems Using Ant Colony Programming, in: Proceedings of AI-METH, pp. 55–60, 2003.
  15. N. Kumaresan, P. Balasubramaniam, Singular Optimal Control for Stochastic Linear Quadratic Singular System Using Ant Colony Programming, Int. J. Comput. Math. 87 (2010), 3311–3327. https://doi.org/10.1080/00207160903026634.
  16. M.Z.M. Kamali, N. Kumaresan, K. Ratnavelu, Solving Differential Equations with Ant Colony Programming, Appl. Math. Model. 39 (2015), 3150–3163. https://doi.org/10.1016/j.apm.2014.11.003.
  17. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer Berlin Heidelberg, 1992. https://doi.org/10.1007/978-3-662-12616-5.
  18. K. Burrage, P.M. Burrage, T. Tian, Numerical Methods for Strong Solutions of Stochastic Differential Equations: An Overview, Proc. R. Soc. Lond. A. 460 (2004), 373–402. https://doi.org/10.1098/rspa.2003.1247.
  19. E. Platen, N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-13694-8.
  20. M. Boryczka, Ant Colony Programming: Application of Ant Colony System to Function Approximation, in: R. Chiong (Ed.), Intelligent Systems for Automated Learning and Adaptation, IGI Global, 2010: pp. 248–272. https://doi.org/10.4018/978-1-60566-798-0.ch011.
  21. M. Boryczka, Ant Colony Programming for Approximation Problems, in: M.A. Kłopotek, S.T. Wierzchoń, M. Michalewicz (Eds.), Intelligent Information Systems 2002, Physica-Verlag HD, Heidelberg, 2002: pp. 147–156. https://doi.org/10.1007/978-3-7908-1777-5_15.
  22. Y. Boudouaoui, H. Habbi, C. Ozturk, D. Karaboga, Solving Differential Equations with Artificial Bee Colony Programming, Soft Comput. 24 (2020), 17991–18007. https://doi.org/10.1007/s00500-020-05051-y.