August  2013, 18(6): 1697-1714. doi: 10.3934/dcdsb.2013.18.1697

Khasminskii-type theorems for stochastic functional differential equations

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

Department of Applied Mathematics, Donghua Univerisity, Shanghai 201600, China

3. 

Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH

4. 

School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China

Received  October 2011 Revised  December 2011 Published  March 2013

For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions.
Citation: Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697
References:
[1]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, J. Math. Anal. Appl., 292 (2004), 364. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics,, International J. Pure and Applied Math., 11 (2004), 377. Google Scholar

[3]

T. C. Gard, "Introduction to Stochastic Differential Equations,", Monographs and Textbooks in Pure and Applied Mathematics, 114 (1988). Google Scholar

[4]

R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations,", Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7 (1980). Google Scholar

[5]

V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of Functional-Differential Equations,", Mathematics and its Applications (Soviet Series), 85 (1992). Google Scholar

[6]

G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities,", Mathematics in Science and Engineering, 150 (1980). Google Scholar

[7]

M. Loève, "Probability Theory,", Third edition, (1963). Google Scholar

[8]

Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations,, Automatica J. IFAC, 47 (2011), 2075. doi: 10.1016/j.automatica.2011.06.014. Google Scholar

[9]

X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales,", Pitman Research Notes in Mathematics Series, 251 (1991). Google Scholar

[10]

X. Mao, "Exponential Stability of Stochastic Differential Equations,", Monographs and Textbooks in Pure and Applied Mathematics, 182 (1994). Google Scholar

[11]

X. Mao, "Stochastic Differential Equations and Applications,", Second edition, (2008). Google Scholar

[12]

X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations,, J. Math. Anal. Appl., 268 (2002), 125. doi: 10.1006/jmaa.2001.7803. Google Scholar

[13]

X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations,, J. Sto. Anal. Appl., 23 (2005), 1045. doi: 10.1080/07362990500118637. Google Scholar

[14]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,", Imperial College Press, (2006). Google Scholar

[15]

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics,, J. Math. Anal. Appl., 304 (2005), 296. doi: 10.1016/j.jmaa.2004.09.027. Google Scholar

[16]

S. E. A. Mohammed, "Stochastic Functional Differential Equations,", Research Notes in Mathematics, 99 (1984). Google Scholar

show all references

References:
[1]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, J. Math. Anal. Appl., 292 (2004), 364. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics,, International J. Pure and Applied Math., 11 (2004), 377. Google Scholar

[3]

T. C. Gard, "Introduction to Stochastic Differential Equations,", Monographs and Textbooks in Pure and Applied Mathematics, 114 (1988). Google Scholar

[4]

R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations,", Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7 (1980). Google Scholar

[5]

V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of Functional-Differential Equations,", Mathematics and its Applications (Soviet Series), 85 (1992). Google Scholar

[6]

G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities,", Mathematics in Science and Engineering, 150 (1980). Google Scholar

[7]

M. Loève, "Probability Theory,", Third edition, (1963). Google Scholar

[8]

Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations,, Automatica J. IFAC, 47 (2011), 2075. doi: 10.1016/j.automatica.2011.06.014. Google Scholar

[9]

X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales,", Pitman Research Notes in Mathematics Series, 251 (1991). Google Scholar

[10]

X. Mao, "Exponential Stability of Stochastic Differential Equations,", Monographs and Textbooks in Pure and Applied Mathematics, 182 (1994). Google Scholar

[11]

X. Mao, "Stochastic Differential Equations and Applications,", Second edition, (2008). Google Scholar

[12]

X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations,, J. Math. Anal. Appl., 268 (2002), 125. doi: 10.1006/jmaa.2001.7803. Google Scholar

[13]

X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations,, J. Sto. Anal. Appl., 23 (2005), 1045. doi: 10.1080/07362990500118637. Google Scholar

[14]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,", Imperial College Press, (2006). Google Scholar

[15]

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics,, J. Math. Anal. Appl., 304 (2005), 296. doi: 10.1016/j.jmaa.2004.09.027. Google Scholar

[16]

S. E. A. Mohammed, "Stochastic Functional Differential Equations,", Research Notes in Mathematics, 99 (1984). Google Scholar

[1]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[2]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015

[3]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[4]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

[5]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[6]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084

[7]

Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885

[8]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065

[9]

Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2019182

[10]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199

[11]

Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287

[12]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[13]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255

[14]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

[15]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[16]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

[17]

Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763

[18]

Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191

[19]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[20]

Jun Zhou, Jun Shen, Weinian Zhang. A powered Gronwall-type inequality and applications to stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7207-7234. doi: 10.3934/dcds.2016114

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (8)

[Back to Top]