Application of an indicator random process for modeling open stochastic systems
https://doi.org/10.25587/2411-9326-2024-2-81-98
Аннотация
The authors present a method of indicator random processes, applicable to constructing models of jump processes associated with the diffusion process. Indicator random processes are processes that take only two values: 1 and 0, in accordance with some probabilistic laws. It is shown that the indicator random process is invariant when reduced to an arbitrary positive degree. Equations with random coefficients used in modeling dynamic systems, when applying the method of indicator random processes, can take into account the possibility of adaptation to external changes, including random ones, in order to preserve indicators important for the existence of the system, which can be continuous or discrete. In the case of indicator random processes, defined as functions of the Poisson process, equations for dynamic processes in a media with abruptly changing properties are constructed and studied. To study the capabilities of the proposed method, dynamic models of the diffusion process in media were studied with delay centers and diffusion processes during transitions by switching from one subspace to another. For these models, equations for characteristic functions are constructed. Using the method of indicator random processes, a characteristic function for the Kac model was constructed. It is shown that in the case of dependence of the indicator random process on the Poisson process, the equation for the characteristic function corresponds to the telegraph equation. This result coincides with the result of Kac.
Об авторах
V. Doobko
Research Center of the Institute of Cybernetics
Украина
Украина
Valery Doobko
Kiev
E. Karachanskaya
Far Eastern State Transport University
Россия
Россия
Elena Karachanskaya
47 Seryshev Street, 680000 Khabarovsk
Список литературы
1. Willems J. C., “Open stochastic systems,” IEEE Trans. Autom. Control, 58, 406–421 (2013).
2. Engel A. and Parker S. E., “Correspondence between open bosonic systems and stochastic differential equations,” Eur. Phys. J. Plus., 138, 578 (2023).
3. Antwi O. A., Bright K. P., and Wereko K. A., “Jump diffusion simulation of stock prices on Ghana stock exchange,” J. Appl. Math. Phys., 8, 1736–1754 (2020).
4. Akyildirim E., Hekimoglu A. A., Sensoy A., et al., “Extending the Merton model with applications to credit value adjustment,” Ann. Oper. Res., 326, 27–65 (2023).
5. Vladimirov I. G., “A transverse Hamiltonian approach to infinitesimal perturbation analysis of quantum stochastic systems,” Entropy, 25, 1179 (2023).
6. Li M. J., “Numerical methods for discrete double barrier option pricing based on Merton jump diffusion model,” Open J. Stat., 7, 446–458 (2017).
7. Novat K., Charles W. M., and Masanja V. G., “Merton’s jump diffusion model an application to stock markets of East African countries,” Int. J. Adv. Sci. Res. Eng., 8, 16–24 (2019).
8. Kou S. G., “Jump-diffusion models for asset pricing,” in: Financial Engineering (J. R. Birge, V. Linetsky, eds.), pp. 73–116, Elsevier (2007) (Handb. Oper. Res. Manage. Sci.; vol. 15).
9. Siu T. K. and Elliott R. J., “American option pricing and filtering with a hidden regimeswitching jump diffusion,” J. Deriv., 29, 106–123 (2022).
10. Ramponi A., “Spread option pricing in regime-switching jump diffusion models,” Math., 10, 1574 (2022).
11. Tran Ky Q., Nguyen D. H., and Yin G., “Stability in distribution and stabilization of switching jump diffusions,” ESAIM: COCV, 28, 72 (2022).
12. Ji H., Shao J., and Xi F., “Stability of regime-switching jump diffusion processes,” J. Math. Anal. Appl., 484, No. 1, 123727 (2020).
13. Jakubowski J. and Niewgowski M., “Jump-diffusion processes in random environments,” J. Differ. Equ., 257, 2671–2703 (2014).
14. Averina T. A. and Rybakov K. A., “Solving approximately a prediction problem for stochastic jump-diffusion systems,” Numer. Anal. Appl., 10, 1–10 (2017).
15. Averina T. A., Karachanskaya E. V., and Rybakov K. A., “Statistical analysis of diffusion systems with invariants,” Russ. J. Numer. Anal. Math. Model., 33, 1–13 (2018).
16. Doobko V. A., “Simulation of distribution of impurities in the environment with delay centers using indicator functions [in Ukrainian],” in: Proc. Agrarian Science and Education in the 21st Century: Problems, Prospects and Innovations (Ukraine, May 17–18, 2018).
17. Doobko V. and Karachanskaya E., “Indicator random processes and its application for modeling open stochastic systems,” https://arxiv.org/pdf/2402.02493.pdf.
18. Klyatskin V. I., Lectures on Dynamics of Stochastic Systems, Elsevier, Burlington, USA (2010).
19. Karachanskaya E. V., Simulation Processes with Invariants [in Russian], Dal’nevost. Nats. Univ., Khabarovsk (2014).
20. Kac M., Probability and Related Topics in Physical Sciences, Intersci. Publ., London (1959).
21. Gihman I. and Skorohod A., Stochastic Differential Equations, Springer, Berlin; New York (1972).
22. Beck J. and Bonetto F., “Grand canonical evolution for the Kac model,” J. Stat. Phys., 188, 7 (2022).
23. Kamke E., Differential Equations, Solution Methods and Solutions, vol. 1: Ordinary Differential Equations, Akad. Verlagsges. Becker & Erler, Leipzig (1951).
Рецензия
Для цитирования:
Doobko V., Karachanskaya E. Application of an indicator random process for modeling open stochastic systems. Математические заметки СВФУ. 2024;31(2):81-98. https://doi.org/10.25587/2411-9326-2024-2-81-98
For citation:
Doobko V., Karachanskaya E. Application of an indicator random process for modeling open stochastic systems. Mathematical notes of NEFU. 2024;31(2):81-98. https://doi.org/10.25587/2411-9326-2024-2-81-98
Просмотров:
3
JATS XML
Послать статью по эл. почте 