# On a comparison theorem for stochastic integro-functional equations of neutral type

### Abstract

In this paper, we will discuss a comparison result for solutions to the Cauchy problems for two stochastic differential equations with delay. On this subject number of authors have obtained their comparison results. We deal with the Cauchy problems for two integro-differential equations. Except transient- (or drift-) and diffusion coefficients our equations include also one integro-differential term. Basic difference of our case from the case of all earlier investigated problems is presence of this term. We introduce a concept of solutions to our problems and prove the comparison theorem for them. According to our result under certain assumptions on coefficients of equations under consideration, their solutions depend on the transient-coefficients in a monotone way.### References

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16. Butkovsky O. "Subgeometric rates of convergence of Markov processes in the Wasserstein metric." The Annals of Applied Probability 24, no. 2 (2014): 526-552.

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21. Es-Sarhir, A., Von Renesse, M.K. and Scheutzow, M. "Harnack inequality for functional SDEs with bounded memory." Electronic communications in probability 14 (2009): 560-565.

22. Es-Sarhir A., Scheutzow M., Tolle J.M. and van Gaans O. "Invariant measures for monotone SPDEs with multiplicative noise term." Applied Mathematics and Optimization 68, no. 2 (2013): 275-287.

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24. Hairer M., Mattingly J.C. and Scheutzow M. "Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations." Probability theory and related fields 149, no. 1-2 (2011): 223-259.

25. Has'minskii R. Z. "Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems." Theory Probability Appl 12 (1967): 144-147.

2. Gal'Cuk L. I. and Davis M. H. A. "A note on a comparison theorem for equations with different diffusions." Stochastics: An International Journal of Probability and Stochastic Processes 6, no. 2 (1982): 147-149.

3. Geib C. and Manthey R. "Comparison theorems for stochastic differential equations in finite and infinite dimensions." Stochastic processes and their applications 53, no. 1 (1994): 23-35.

4. Huang Zhi Yuan. "A comparison theorem for solutions of stochastic differential equations and its applications." Proceedings of the American Mathematical Society 91, no. 4 (1984): 611-617.

5. Kotelenez P. "Comparison methods for a class of function valued stochastic partial differential equations." Probability Theory and related fields 93, no. 1 (1992): 1-19.

6. Manthey R. and Zausinger T. "Stochastic evolution equations in $L^{2\nu}{\rho}$." Stochastics: An International Journal of Probability and Stochastic Processes 66, no. 1-2 (1999): 37-85.

7. O'Brien G. L. "A new comparison theorem for solutions of stochastic differential equations." Stochastics 3, no. 1-4 (1980): 245-249.

8. Ouknine Y. "Comparaison et non-confluence des solutions d’équations différentielles stochastiques unidimensionnelles." Probab. Math. Statist 11 (1990): 37-46.

9. Tanaka H. "Stochastic differential equations" Seminar on Probability (in Japanese) 19 (1964)

10. Yamada T. "On a comparison theorem for solutions of stochastic differential equations and its applications." Journal of Mathematics of Kyoto University 13, no. 3 (1973): 497-512.

11. Yamada T. and Yukio O. "On the strong comparison theorems for solutions of stochastic differential equations." Zeitschrift for Wahrscheinlichkeits theorie und Verwandte Gebiete 56, no. 1 (1981): 3-19.

12. Da Prato Giuseppe and Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Vol. 152. Cambridge University Press, 2014.

13. Watanabe S. and Ikeda, N. {Stokhasticheskie differencial'nye uravneniya i diffusionnye processy} [Stochastic differential equations and diffusional processes].(M.:1986): Nauka, 445.

14. Skorokhod A. V. { Issledovaniya po teorii sluchainyh processov} [Research on the theory of random processes] .(Kiev: Kiev University, 1961):216

15. Stanzhitskii A.N. and Tsukanova A.O. { Sushestvovanie i edinstvennost' resheniya zada Koshi dlya stohasticheskogo differencialnogouravneniya reakcii-diffuzii neitralnogo tipa} [Existence and uniqueness of a solution to the Cauchy problem for a stochastic differential reaction-diffusion equation of neutral type]. Nelineinoe kolebanie. 19,$\No$3 (2016): 408-430.

16. Butkovsky O. "Subgeometric rates of convergence of Markov processes in the Wasserstein metric." The Annals of Applied Probability 24, no. 2 (2014): 526-552.

17. Butkovsky O. and Michael S.. "Invariant measures for stochastic functional differential equations." Electronic Journal of Probability 22 (2017).

18. Butkovsky O., Kulik A. and Scheutzow M.. "Generalized couplings and ergodic rates for SPDEs and other Markov models." arXiv preprint arXiv:1806.00395 (2018).

19. Doblin W. "Elements d'une theorie generale des chaînes simples constantes de Markoff." In Annales Scientifiques de l'ENS, vol. 57, pp. 61-111. 1940.

20. Es-Sarhir, A., van Gaans, O. and Scheutzow, M. "Invariant measures for stochastic functional differential equations with superlinear drift term." Differential and Integral Equations 23, no. 1/2 (2010): 189-200.

21. Es-Sarhir, A., Von Renesse, M.K. and Scheutzow, M. "Harnack inequality for functional SDEs with bounded memory." Electronic communications in probability 14 (2009): 560-565.

22. Es-Sarhir A., Scheutzow M., Tolle J.M. and van Gaans O. "Invariant measures for monotone SPDEs with multiplicative noise term." Applied Mathematics and Optimization 68, no. 2 (2013): 275-287.

23. Hairer M. and Mattingly J.C. "Yet another look at Harris’ ergodic theorem for Markov chains." In Seminar on Stochastic Analysis, Random Fields and Applications VI, pp. 109-117. Springer, Basel, 2011.

24. Hairer M., Mattingly J.C. and Scheutzow M. "Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations." Probability theory and related fields 149, no. 1-2 (2011): 223-259.

25. Has'minskii R. Z. "Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems." Theory Probability Appl 12 (1967): 144-147.

Published

2020-04-05

How to Cite

STANZHITSKII, A. N.; KARAKENOVA, S. G.; ZHUMATOV, S. S..
On a comparison theorem for stochastic integro-functional equations of neutral type.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 105, n. 1, p. 30-45, apr. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/699>. Date accessed: 07 june 2020.
Section

Mathematics

Keywords
stochastic differential equation, comparison theorem, Hilbert space