Geodesic
On~stability and comparison theorems for~systems of~stochastic differential equations
Matematičeskie trudy, Tome 22 (2019) no. 1, pp. 3-18.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove comparison theorems for stochastic differential equations (briefly, SDE) with respect to a standard multidimensional Wiener process as well as for components of systems of SDE with respect to a multidimensional Wiener process. The obtained results are applied to the study of the stability with probability 1 of the perturbed solutions to the SDE. 
@article{MT_2019_22_1_a0,
     author = {A. S. Asylgareev},
     title = {On~stability and comparison theorems for~systems of~stochastic differential equations},
     journal = {Matemati\v{c}eskie trudy},
     pages = {3--18},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2019_22_1_a0/}
}
TY  - JOUR
AU  - A. S. Asylgareev
TI  - On~stability and comparison theorems for~systems of~stochastic differential equations
JO  - Matematičeskie trudy
PY  - 2019
SP  - 3
EP  - 18
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2019_22_1_a0/
LA  - ru
ID  - MT_2019_22_1_a0
ER  - 
%0 Journal Article
%A A. S. Asylgareev
%T On~stability and comparison theorems for~systems of~stochastic differential equations
%J Matematičeskie trudy
%D 2019
%P 3-18
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2019_22_1_a0/
%G ru
%F MT_2019_22_1_a0
A. S. Asylgareev. On~stability and comparison theorems for~systems of~stochastic differential equations. Matematičeskie trudy, Tome 22 (2019) no. 1, pp. 3-18. http://geodesic.mathdoc.fr/item/MT_2019_22_1_a0/

[1] Asylgareev A. S., Nasyrov F. S., “O teoremakh sravneniya i ustoichivosti s veroyatnostyu 1 odnomernykh stokhasticheskikh differentsialnykh uravnenii”, Sib. matem. zhurn., 57:5 (2016), 969–977 | MR | Zbl

[2] Vatanabe S., Ikeda N., Stokhasticheskie differentsialnye uravneniya i diffuzionnye protsessy, Nauka, M., 1986

[3] Krasnoselskii M. A., Pokrovskii A. V., “Estestvennye resheniya stokhasticheskikh differentsialnykh uravnenii”, Dokl. AN SSSR, 240:2 (1978), 264–267 | MR

[4] Kuznetsov D. F., Chislennoe modelirovanie stokhasticheskikh differentsialnykh uravnenii i stokhasticheskikh integralov, Nauka, SPb, 1999

[5] Kushner G. Dzh., Stokhasticheskaya ustoichivost i upravlenie, Mir, M., 1969

[6] Nasyrov F. S., Lokalnye vremena, simmetrichnye integraly i stokhasticheskii analiz, Fizmatlit, M., 2011

[7] Nasyrov F. S., “Ob integrirovanii sistem stokhasticheskikh differentsialnykh uravnenii”, Matem. tr., 19:2 (2016), 158–169 | Zbl

[8] Skorokhod A. V., Issledovaniya po teorii sluchainykh protsessov, Izd-vo Kievskogo un-ta, Kiev, 1961

[9] Khasminskii R. Z., Ustoichivost sistem differentsialnykh uravnenii pri sluchainykh vozmuscheniyakh ikh parametrov, Nauka, M., 1969

[10] Doss H., “Connections between stochastic and ordinary integral equations”, Biological Growth and Spread, Mathematical Theories and Applications, Proc. Conf. (Heidelberg, 1979), Lect. Notes Biomath., 38, 1980, 443–448 | DOI | MR | Zbl

[11] Geiß C., Manthey R., “Comparison theorems for stochastic differential equations in finite and infinite dimensions”, Stochastic Processes Appl., 53:1 (1994), 23–35 | DOI | MR | Zbl

[12] Mao X., Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc, New York, 1994 | MR | Zbl

[13] O'Brien G. L., “A new comparison theorem for solutions of stochastic differential equations”, Stochastics, 3:4 (1980), 245–249 | DOI | MR | Zbl