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November  2019, 24(11): 5927-5944. doi: 10.3934/dcdsb.2019113

Almost periodic solutions and stable solutions for stochastic differential equations

Yong Li 1,2, , Zhenxin Liu 3, and  Wenhe Wang 4,,

1. 

School of Mathematics, Jilin University, Changchun 130012, China
2. 

Center for Mathematics and Interdisciplinary Sciences, and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
3. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
4. 

School of Basic Sciences, Changchun University of Technology, Changchun 130012, China

* Corresponding author

Received  September 2018 Published  November 2019 Early access  June 2019

Fund Project: The first author is supported by National Research Program of China Grant 2013CB834100, Jilin DRC Grant 2017C028-1, NSFC Grants 11571065 and 11171132; the second author is supported by NSFC Grants 11522104 and 11871132, and the startup and Xinghai Jieqing funds from Dalian University of Technology.

In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Our essential idea is to get stability of solutions or systems by some inherited properties of Lyapunov functions. Under suitable conditions besides Lyapunov functions, we obtain the existence of almost periodic solutions in distribution.

Keywords: Stochastic differential equation, almost periodic solution, asymptotically almost periodic solution, stable in distribution, Lyapunov function.
Mathematics Subject Classification: Primary: 60H10, 34C27; Secondary: 37B25.
Citation: Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
References:
[1]

L. Amerio and G. Prouse, Almost-periodic Functions and Functional Equations, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne, 1971. viii+184 pp.

[2]

L. Arnold and C. Tudor, Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics Stochastics Rep., 64 (1998), 177-193.  doi: 10.1080/17442509808834163.

[3]

G. K. Basak, A class of limit theorems for singular diffusions, J. Multivariate Anal., 39 (1991), 44-59.  doi: 10.1016/0047-259X(91)90004-L.

[4]

G. K. Basak and R. N. Bhattachaya, Stability in distribution for a class of singular diffusions, Ann. Probab., 20 (1992), 312-321.  doi: 10.1214/aop/1176989928.

[5]

J. E. Bertram and P. E. Sarachik, Stability of circuits with randomly time-varying parameters, IRE Trans. Information Theory, 6 (1959), 260-270. 

[6]

S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, I. Funktionen einer Variablen, (German), Math. Ann., 96 (1927), 119-147.  doi: 10.1007/BF01209156.

[7]

S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039.

[8]

H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅰ., Acta Math., 45 (1925), 29-127.  doi: 10.1007/BF02395468.

[9]

H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅱ., Acta Math., 46 (1925), 101-214.  doi: 10.1007/BF02543859.

[10]

H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅲ., Acta Math., 47 (1926), 237-281.  doi: 10.1007/BF02543846.

[11]

K. L. Chung, A Course in Probability Theory, 3nd edition, Academic press, Inc., San Diego, CA, 2001. xviii+419 pp.

[12]

W. A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Mat. Pura Appl.(4), 76 (1967), 27-49.  doi: 10.1007/BF02412227.

[13]

G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380.

[14]

L. G. Deysach and G. R. Sell, On the existence of almost periodic motion, Michgan Math. J., 12 (1965), 87-95.  doi: 10.1307/mmj/1028999248.

[15]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer-Verlag, Berlin-New York, 1974. viii+336 pp.

[16]

A. Halanay, Periodic and almost periodic solutions to affine stochastic systems, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations, (Budapest, 1987), 94–101, János Bolyai Math. Soc., Budapest, 1987.

[17]

J. Hale, Periodic and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 15 (1964), 289-304.  doi: 10.1007/BF00249199.

[18]

I. Ya. Kac and N. N. Krasovskii, On the stability of systems with random parameters, (Russian), J. Appl. Math. Mech., 24 (1960), 1225-1246. 

[19]

R. Khasminskii, On the stability of the trajectory of Markov processes,, J. Appl. Math. Mech., 26 (1962), 1554-1565.  doi: 10.1016/0021-8928(62)90192-2.

[20]

R. Khasminskii, Stochastic Stability of Differential Equations, 1$^{nd}$ edition, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980. xvi+344 pp; 2$^{nd}$ edition, Springer, Heidelberg, 2012. xviii+339 pp.

[21]

F. Kozin, On almost sure stability of linear systems with random coefficients, J. Math. Phys., 42 (1963), 59-67.  doi: 10.1002/sapm196342159.

[22]

H. J. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Vol. 33 Academic Press, New York-London, 1967. xiv+161 pp.

[23]

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Translated from the Russian by L. W. Longdon, Cambridge University Press, CambridgeNew York, 1982. xi+211 pp.

[24]

Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019.

[25]

X. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994. xii+307 pp.

[26]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics and Applications, Horwood Publishing Limited, Chichester, 1997. xii+366 pp.

[27]

A. A. Markov, Stabilität in Liapvanoffschen Sinne und Fastperiodizität, Math. Z., 36 (1933), 708-738.  doi: 10.1007/BF01188645.

[28]

R. K. Miller, Almost periodic differential equations as dynamical systems with applications to the extence of a. p. solution, J. Differential Equations, 1 (1965), 337-345.  doi: 10.1016/0022-0396(65)90012-4.

[29]

T. Morozan and C. Tudor, Almost periodic solutions of affine Itô equations, Stoch. Anal. Appl., 7 (1989), 451-474.  doi: 10.1080/07362998908809194.

[30]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London, 1967. xi+276 pp.

[31]

G. Seifert, Almost periodic solutions for almost periodic systems of ordinary differential equations, J. Differential Equations, 2 (1966), 305-319.  doi: 10.1016/0022-0396(66)90071-4.

[32]

G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262.  doi: 10.2307/1994645.

[33]

G. R. Sell, Nonautonomous differential equations and topological dynamics. Ⅱ. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283.  doi: 10.1090/S0002-9947-1967-0212314-4.

[34]

C. Vârsana, Asymptotic almost periodic solutions for stochastic differential equations, Tôhoku Math. J., 41 (1989), 609-618.  doi: 10.2748/tmj/1178227731.

[35]

T. Yoshizawa, Extreme stability and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 17 (1964), 148-170.  doi: 10.1007/BF00253052.

[36]

T. Yoshizawa, Asymptotically almost periodic solutions of an almost periodic system, Funkcial. Ekvac., 12 (1969), 23-40. 

[37]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp.

show all references

References:
[1]

L. Amerio and G. Prouse, Almost-periodic Functions and Functional Equations, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne, 1971. viii+184 pp.

[2]

L. Arnold and C. Tudor, Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics Stochastics Rep., 64 (1998), 177-193.  doi: 10.1080/17442509808834163.

[3]

G. K. Basak, A class of limit theorems for singular diffusions, J. Multivariate Anal., 39 (1991), 44-59.  doi: 10.1016/0047-259X(91)90004-L.

[4]

G. K. Basak and R. N. Bhattachaya, Stability in distribution for a class of singular diffusions, Ann. Probab., 20 (1992), 312-321.  doi: 10.1214/aop/1176989928.

[5]

J. E. Bertram and P. E. Sarachik, Stability of circuits with randomly time-varying parameters, IRE Trans. Information Theory, 6 (1959), 260-270. 

[6]

S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, I. Funktionen einer Variablen, (German), Math. Ann., 96 (1927), 119-147.  doi: 10.1007/BF01209156.

[7]

S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039.

[8]

H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅰ., Acta Math., 45 (1925), 29-127.  doi: 10.1007/BF02395468.

[9]

H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅱ., Acta Math., 46 (1925), 101-214.  doi: 10.1007/BF02543859.

[10]

H. Bohr, Zur theorie der fastperiodischen funktionen, (German) Ⅲ., Acta Math., 47 (1926), 237-281.  doi: 10.1007/BF02543846.

[11]

K. L. Chung, A Course in Probability Theory, 3nd edition, Academic press, Inc., San Diego, CA, 2001. xviii+419 pp.

[12]

W. A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Mat. Pura Appl.(4), 76 (1967), 27-49.  doi: 10.1007/BF02412227.

[13]

G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380.

[14]

L. G. Deysach and G. R. Sell, On the existence of almost periodic motion, Michgan Math. J., 12 (1965), 87-95.  doi: 10.1307/mmj/1028999248.

[15]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377. Springer-Verlag, Berlin-New York, 1974. viii+336 pp.

[16]

A. Halanay, Periodic and almost periodic solutions to affine stochastic systems, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations, (Budapest, 1987), 94–101, János Bolyai Math. Soc., Budapest, 1987.

[17]

J. Hale, Periodic and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 15 (1964), 289-304.  doi: 10.1007/BF00249199.

[18]

I. Ya. Kac and N. N. Krasovskii, On the stability of systems with random parameters, (Russian), J. Appl. Math. Mech., 24 (1960), 1225-1246. 

[19]

R. Khasminskii, On the stability of the trajectory of Markov processes,, J. Appl. Math. Mech., 26 (1962), 1554-1565.  doi: 10.1016/0021-8928(62)90192-2.

[20]

R. Khasminskii, Stochastic Stability of Differential Equations, 1$^{nd}$ edition, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980. xvi+344 pp; 2$^{nd}$ edition, Springer, Heidelberg, 2012. xviii+339 pp.

[21]

F. Kozin, On almost sure stability of linear systems with random coefficients, J. Math. Phys., 42 (1963), 59-67.  doi: 10.1002/sapm196342159.

[22]

H. J. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Vol. 33 Academic Press, New York-London, 1967. xiv+161 pp.

[23]

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Translated from the Russian by L. W. Longdon, Cambridge University Press, CambridgeNew York, 1982. xi+211 pp.

[24]

Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019.

[25]

X. Mao, Exponential Stability of Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994. xii+307 pp.

[26]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics and Applications, Horwood Publishing Limited, Chichester, 1997. xii+366 pp.

[27]

A. A. Markov, Stabilität in Liapvanoffschen Sinne und Fastperiodizität, Math. Z., 36 (1933), 708-738.  doi: 10.1007/BF01188645.

[28]

R. K. Miller, Almost periodic differential equations as dynamical systems with applications to the extence of a. p. solution, J. Differential Equations, 1 (1965), 337-345.  doi: 10.1016/0022-0396(65)90012-4.

[29]

T. Morozan and C. Tudor, Almost periodic solutions of affine Itô equations, Stoch. Anal. Appl., 7 (1989), 451-474.  doi: 10.1080/07362998908809194.

[30]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London, 1967. xi+276 pp.

[31]

G. Seifert, Almost periodic solutions for almost periodic systems of ordinary differential equations, J. Differential Equations, 2 (1966), 305-319.  doi: 10.1016/0022-0396(66)90071-4.

[32]

G. R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc., 127 (1967), 241-262.  doi: 10.2307/1994645.

[33]

G. R. Sell, Nonautonomous differential equations and topological dynamics. Ⅱ. Limiting equations, Trans. Amer. Math. Soc., 127 (1967), 263-283.  doi: 10.1090/S0002-9947-1967-0212314-4.

[34]

C. Vârsana, Asymptotic almost periodic solutions for stochastic differential equations, Tôhoku Math. J., 41 (1989), 609-618.  doi: 10.2748/tmj/1178227731.

[35]

T. Yoshizawa, Extreme stability and almost periodic solutions of functional-differential equations, Arch. Rational Mech. Anal., 17 (1964), 148-170.  doi: 10.1007/BF00253052.

[36]

T. Yoshizawa, Asymptotically almost periodic solutions of an almost periodic system, Funkcial. Ekvac., 12 (1969), 23-40. 

[37]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp.

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