DOI
10.34229/KCA2522-9664.26.4.14
UDC 519.21
S.A. Semenyuk
Lviv Polytechnic National University, Lviv, Ukraine,
serhii.a.semeniuk@lpnu.ua
CONVERGENCE IN PROBABILITY FOR A STOCHASTIC
OPTIMAL CONTROL PROBLEMIN THE AVERAGING SCHEMA
Abstract. The article is devoted to the study of the convergence for the problem of stochastic optimal control with Markov switching in the averaging scheme.
The combination of the standard Wiener process with a uniformly ergodic Markov process is considered, which enables the description of the system’s evolution
under the influence of diffusion noise and random mode switching. Ergodic properties of the fast process and conditions for the regularity of coefficients are provided,
ensuring the system’s stable behavior on average. This allows us to prove the convergence in probability
(through Lp-estimates) of the trajectories of the original system to the solutions of the marginal averaged system. The obtained results enable the simplification of problem studies in stochastic optimization and optimal control.
Keywords: stochastic optimal control, averaging principle, convergence in probability, stochastic differential equations, Wiener process.
full text
REFERENCES
- Dupacova J., Hurt J., Stepan J. Stochastic modeling in economics and finance. Applied Optimization. Kluwer. N.-Y.: Springer, 2002. 386 p. https://doi.org/10.1007/b101992.
- Zhuravel I., Semenyuk S., Korobeinikova T., Romanyuk O., Reshetnik O., Maydanyuk V. Stochastic SIR model for computer worm propagation, 2025 15th International Conference on Advanced Computer Information Technologies (ACIT). Sibenik, Croatia. 2025. P. 505–508. https://doi.org/10.1109/ACIT65614.2025.11185628.
- Samoilenko A.M., Stanzhytskyi O.M. Qualitative and asymptotic analysis of differential equations with random perturbations. Singapore, World Scientific, 2011. 324 p. https://doi.org/10.1142/8016.
- Ankirchner S., Engelhardt S. Long term average cost control problems without ergodicity. Appl. Math. Optim. 2022. Vol. 86, N 42. https://doi.org/10.1007/s00245-022-09902-y.
- Semenyuk S.A., Chabanyuk Y., Chypurko R.A., Lytvyn A.A. Control problem for the Markov-modulated Poisson process in the diffusion schema. Matematychni Studii. 2025. Vol. 64, N 1. P. 99–106. https://doi.org/10.30970/ms.64.1.99-106.
- Chabanyuk Y.M., Semenyuk S.A., Khimka U.T., Chypurko R.A. Stochastic evolution under Markov-modulated Poisson perturbation in the diffusion approximation scheme. Cybern. Syst. Anal. 2025. Vol. 61, N 3. P. 443–449. https://doi.org/10.1007/s10559-025-00781-z.
- Korolyuk V.S., Limnios N., Samoilenko I.V. Levy and Poisson approximations of switched stochastic systems by a semimartingale approach. Comptes Rendus Mathematique. 2016. Vol. 354. P. 723–728. https://doi.org/10.1016/j.crma.2016.04.008.
- Skorokhod A.V., Hoppensteadt F.C., Salehi H. Random perturbation method with application in science and engineering. N.-Y.: Springer, 2002. 490 p. https://doi.org/10.1007/b98905.
- Koroliouk D., Samoilenko I. Asymptotic and analytic methods in stochastic evolutionary symptoms. John Wiley & Sons. 2023. 234 p. https://doi.org/10.1002/9781394229482.
- Chabanyuk Y., Nikitin A., Khimka U. Asymptotic analyses for complex evolutionary systems with Markov and Semi-Markov switching using approximation schemes, mathematics and statistics. John Wiley & Sons, 2020.
- Xiaobin Sun, Longjie Xie, Yingchao Xie. Strong and weak convergence rates for slow–fast stochastic differential equations driven by α-stable process. Bernoulli, 2022. Vol. 28, N 1. P. 343–369. https://doi.org/10.3150/21-BEJ1345.
- Shen Guangjun, Fan Jingjing, Wu Jiang-Lun, Wang Zhi. Stochastic averaging principle for neutral stochastic functional differential equations driven by G-Lvy process. Stochastics and Dynamics. 2024. Vol. 24, N 4. 2450029, https://doi.org/10.1142/S0219493724500291.
- Yanyan Du, Yuming Chen, Qimin Zhang. Averaging principle and optimal control for a two-time-scale stochastic reaction-diffusion HIV model. Adv. Cont. Discr. Mod. 2025. Vol. 177. https://doi.org/10.1186/s13662-025-04036-1.
- Ruomiao Huang, Danfeng Luo. Strong averaging principle for generalized Caputo fractional stochastic neutral differential equations driven by multiplicative fractional Brownian motion. Chaos. Solitons & Fractals. 2025. Vol. 201, N 1. https://doi.org/10.1016/j.chaos.2025.117179.
