Asymptotic behavior of sample paths for retarded stochastic differential equations without dissipativity

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1 Liu and Song Advances in Difference Equations 15) 15:177 DOI /s R E S E A R C H Oen Access Asymtotic behavior of samle aths for retarded stochastic differential equations without dissiativity Zaiming Liu and Na Song * * Corresondence: songna@csu.edu.cn School of Mathematics and Statistics, Central South University, Changsha, 4183, China Abstract This aer is concerned with two classes of retarded stochastic differential equations. Sufficient conditions are derived to guarantee the th moment exonential stability and almost sure exonential stability. Moreover, we construct some examles to demonstrate the theory derived. Keywords: retarded stochastic differential equation; variation-of-constants formula; neutral tye equation; exonential stability 1 Introduction The theory of retarded stochastic differential equations SDEs) has received a great deal of attention since it is not only academically challenging but also of ractical imortance, and it has layed an imortant role in many ways such as in life insurance, risk management, wireless communication, and otimal control of multiagent systems. Retarded stochastic differential equations SDEs) are such SDEs that involve retarded arguments. There is extensive literature on the stability of stochastic differential delay equations see, e.g.,[1 7]). In the ast few decades, the theory of neutral stochastic differential equations has also received a great deal of attention. Mao et al. [8] studied the almost sure asymtotic stability of the neutral stochastic differential delay equations with Markovian switching, while Bao et al. [9] investigated the stability in distribution of neutral stochastic differential delay equations with Markovian switching. Wu et al. [1] studied the mean square asymtotic stability of a generalized half-linear neutral stochastic differential equation with variable delays alying fixed oint theory. Under a non-lischitz condition and a weakened linear growth condition, Bao et al. [11] investigated the existence and uniqueness of mild solutions to stochastic neutral artial functional differential equations. Milošević [1]not only considered global almost sure asymtotic exonential stability of the equilibrium solution for a class of neutral stochastic differential equations with time-deendent delay under nonlinear growth conditions but also established moment estimates for solutions of equations of this tye. Bao et al. [13] used a variation-of-constants formula to overcome the difficulties due to the lack of information at the current time and established existence and uniqueness of stationary distributions for retarded SDEs that need not satisfy dissiative conditions. For more details, we refer to [14 17]. 15 Liu and Song. This article is distributed under the terms of the Creative Commons Attribution 4. International License htt://creativecommons.org/licenses/by/4./), which ermits unrestricted use, distribution, and reroduction in any medium, rovided you give aroriate credit to the original authors) and the source, rovide a link to the Creative Commons license, and indicate if changes were made.

2 Liuand SongAdvances in Difference Equations 15) 15:177 Page of 1 Borrowing the idea of Dung [18], in this aer our investigation is focused on asymtotic behavior of the solution for two classes of retarded SDEs, i.e., semi-linear retarded SDEs and semi-linear retarded SDEs of neutral tye. Sufficient conditions are derived to guarantee the th moment exonential stability and almost sure exonential stability. So far there have been few results resented on the exonential asymtotic behavior of solutions of semi-linear retarded stochastic differential equations. The main aim of this work is to close this ga. The remainder of this aer roceeds as follows. Section rovides some basic notation that will be used in the forthcoming sections and then reviews the variation-of-constants formula for deterministic linear retarded systems. Section 3 is devoted to the study of asymtotic behavior of the solution for semi-linear retarded SDEs and derived sufficient conditions to guarantee the th moment exonential stability and almost sure exonential stability. Section 4 generalizes the theoryestablished in Section3 to semi-linear retarded SDEs of neutral tye. Preliminaries Let, P, F ) be a robability sace together with a filtration {F t } t satisfying the usual conditions i.e., F t+ := s>t F s = F t, F s F t for s t, andf contains all P-null sets). Let {Wt)} t be a real-valued Brownian motion defined on the stochastic basis, P, F, {F t } t ). Fix τ, ), which is referred to as the delay. var denotes the total variation. Use μ ) andρ ) to denote the finite signed measures defined on [,]. Let C be the set of all comlex numbers and Rez) stand for the real art of z C. Throughout this aer, c > is used as a generic ositive constant whose values may change for different usage. Recall the following deterministic linear retarded equation: dy t)= Y t + θ)μdθ) dt 1) with the initial value Y t)=ξt), t [,]. Define Zt) as the fundamental solution of equation 1) with the initial value Z) = 1 and Zθ)=forθ [, ]. By the variation-of-constants formula see, e.g.,[19], Theorem 1.,.17), the solution of 1)hastheuniqueexlicitform Y t; ξ)=zt)ξ) + Let θ Zt + θ s)ξs) dsμdθ), t. ) υ = su { Reλ):λ C, λ)= }, where λ):=λ e λs μds), λ C.

3 Liuand SongAdvances in Difference Equations 15) 15:177 Page 3 of 1 λ) = is called the characteristic equation of equation 1) see,e.g., [19]). Then, according to [19], Theorem 3.,.71), for any α > υ,thereexistsc α >suchthat Zt) cα e αt, t. 3) Before the end of this section, we first give the following lemma which will be a crucial tool for roofs of our main results. Lemma 1 [18], Lemma 3.1) Suose that f CR +, R + ) L 1 R + ) satisfies f t)e γ t dt < for some γ >. 4) If λ >and λ = λ γ, then e λt s) f s) ds <e λ t f s)e γ s ds. 5) 3 Asymtotic behavior of the solution for retarded SDE driven by a Brownian motion In this section, we consider a retarded SDE in the form ) dxt)= Xt + θ)μdθ) dt + σ t) dwt) 6) with the initial value Xt)=ξt), t [,]. Theorem 1 [], Theorem 3.1) There is a unique strong solution {Xt), t } of equation 6), and the solution can be reresented exlicitly by Xt)=Zt)ξ) + Zt + θ s)ξs) dsμdθ)+ θ Zt s)σ s) dws), 7) in which {Zt), t } is the fundamental solution of equation 1) with the initial value Z) = 1 and Zθ)=for θ [,]. The following two theorems rovide the th moment exonential stability and almost sure exonential stability of the solution of equation 6). Theorem Assume that υ <,and there exists γ >such that e γ s σ s) ds <. 8) Then there exist ositive constants C and κ such that, for each >,the solution of equation 6) satisfies E Xt) C e κt, t. 9)

4 Liuand SongAdvances in Difference Equations 15) 15:177 Page 4 of 1 Proof According to the exlicit form 7), along with the following inequality, x, y, z >, x + y + z) c x + y + z ), 1) where c =1if< 1andc =3 1 if >1,itiseasytoget E Xt) = E Zt)ξ) + Zt + θ s)ξs) dsμdθ)+ Zt s)σ s) dws) θ c E Zt)ξ) + E Zt + θ s)ξs) dsμdθ) θ ) + E Zt s)σ s) dws) =: c I1 t)+i t)+i 3 t) ). We shall consider I i t), i =1,,3,resectively.ForI 1 t), by inequality 3), it is obvious that for some α >, I 1 t) c α e αt ξ. 11) For I t), by Hölder s inequality and inequality 3), I t) E Zt + θ s) ξs) ds μ var dθ) θ [ E Zt + θ s) ξs) ) 1 ) 1 q ] ds) μ var dθ) μ var dθ) θ [ ) ) ] μ var )) = E Zt + θ s) ξs) 1 ds μ var dθ) [,] θ [ E θ 1 Zt + θ s) ) ] ξs) μ var )) 1 ds μ var dθ) [,] θ c α e αt ξ )) 1 μ var [,] θ 1 e αθ s) ds μ var dθ) 1/)) c α eατ ξ μ var [,] )) e αt, 1) θ where 1/ +1/q =1,, q [1, ). For I 3 t), by Itô isometry formula, we have E Zt s)σ s) dws) = Let λ 1 = minα, γ )andλ = λ 1. By Lemma 1, E Zt s)σ s) dws) c α e λ 1t where c 1 := c α e λ 1t σ s) e γ t ds. t Zt s)σ s) ds c α e αt s) σ s) ds. 13) σ s) e γ t ds = c 1 e λ t, 14)

5 Liuand SongAdvances in Difference Equations 15) 15:177 Page 5 of 1 Then, by virtue of the Gaussian roerty, for any >, E S = / Ɣ +1 ) Ɣ 1 ) a, 15) where S denotes a random variable following an N, a )law.itisobviousthat I 3 t) / Ɣ +1 ) Ɣ 1 ) c 1 e λ t. 16) Then, by combining 11), 1)and16), we conclude that E Xt) c c α e αt ξ + 1/)) c α eατ ξ )) e μ var [,] αt + / Ɣ +1 ) ) Ɣ 1 ) c 1 e λ t C e κt, where κ := minα, λ )and C = c c α ξ + 1/)) c α eατ ξ )) / Ɣ +1 μ var [,] + ) ) Ɣ 1 ) c 1 <. This comletes the roof. Theorem 3 Assume that υ <and that 8) is satisfied. Then the solution of 6) is almost surely exonentially stable, i.e., there exists β >such that lim su t 1 t log Xt) β, a.s. 17) Proof Note that for n 1 t n, n 1, Xt)canbereresentedas n ) Xt)=Xn 1)+ Xs + θ)ρdθ) ds + σ s) dws). 18) With the inequality a 1 + a + a 3 3 a 1 + a + a 3 ), it follows that E su t n ) Xt) 3 E n Xn 1) + E + E su t n )) σ s) dws) ) ) Xs + θ)ρdθ) ds =: 3 J 1 t)+j t)+j 3 t) ). 19) We shall study J i t)fori =1,,3,resectively.ForJ 1 t), by inequality 9), it is obvious that J 1 t) C e κ). )

6 Liuand SongAdvances in Difference Equations 15) 15:177 Page 6 of 1 For J t), by Hölder s inequality and inequality 9), we have n ) ) J t) E Xs + θ) ρ var dθ) ds [ n ) 1 n E ds Xs + θ) ) 1 ] ρ var dθ)) ds n ) = E Xs + θ) ρ var dθ)) ds { n [ ) 1 1 E ρ var dθ) Xs + θ) ] } ρ var dθ)) ds = ρ var [,] ) n ρ var [,] ) n E Xs + θ) ρ var dθ) ds C e κs+θ) ρ var dθ) ds C ρ var [,] )) e κτ e κ). 1) For J 3 t), by the Burkholder-Davis-Gundy inequality see [1], Theorem 7.3,.4), there exist c >andλ 3 such that J 3 t) c n σ s) ds c e λ 3) where c 3 := c σ s) e λ 3s ds <. Then, by combining )-), we conclude that E su t n Xt) ) σ s) e λ 3 s ds c 3 e λ3), ) 3 C e κ) + C ρ var [,] )) e κτ e κ) + c 3 e λ 3) ) c 4 e λ 4n, 3) where λ 4 := minκ,λ 3 )and c 4 =3 C e κ + C ρ var [,] )) e κτ e κ + c 3 e λ 3 ). Then, with the Chebyshev inequality, for any λ 5 < λ 4, P su t n ) Xt) >e λ 5 n e λ5n E su t n ) Xt) c 4 e λ 4 λ 5 )n. 4) Since n=1 e λ 4 λ 5 )n <, by virtue of Borel-Cantelli lemma see [1], Lemma.4,.7), there exists with P ) = 1 such that for any ω there exists an integer n ω), when n n ω)andn 1 t n, Xt) e λ 5 n e λ 5t. 5) The desired conclusion 17)issatisfiedwithβ = λ 5.

7 Liuand SongAdvances in Difference Equations 15) 15:177 Page 7 of 1 Examle 1 Consider the semi-linear retarded SDE dxt)= Xt 1)dt + σ t) dwt), Xt)=ξt). 6) It is easy to see that the corresonding characteristic equation of 6)is λ +e λ =. 7) A simle calculation using Matlab yields that the unique root of 7) isλ = i. Thus, from this condition, together with 8), we deduce that the solution of 6) is th moment exonentially stable and almost surely exonentially stable by Theorem and Theorem 3. 4 Extension to retarded SDE of neutral tye In this section, we roceed to generalize Theorem and Theorem 3 to SDEs of neutral tye. To begin, we give an overview of the variation-of-constants formula for linear equations of neutral tye. Recall the following deterministic linear retarded equation of neutral tye: d Y t) ) ) Y t + θ)ρdθ) = Y t + θ)μdθ) dt 8) with the initial value Y t)=ξt), t [,]. By [19], Theorem 1.1,.56), equation 8)hasauniquesolution{Yt), t }.Define Zt) as the fundamental solution of equation 8) with the initial value Z) = 1 and Zθ)= forθ [, ]. Then, for any ξ C such that ξ θ) dθ <, Y t) canbeexressed exlicitly by Y t; ξ)=zt)ξ) + Let θ Zt + θ)ξ)ρdθ)+ θ Zt + θ s)ξs) dsμdθ) Zt s + θ)ξ s) dsρdθ). 9) ῡ = su { Reλ):λ C, h λ)= }, where h λ):=λ λ e λθ ρdθ) e λθ μdθ), λ C. Then, according to [19], Theorem 3.,.71), for any ᾱ > ῡ,thereexistscᾱ >such that Zt) cᾱeᾱt, t. 3)

8 Liuand SongAdvances in Difference Equations 15) 15:177 Page 8 of 1 In this section, we consider a retarded SDE of neutral tye in the form d Xt) ) ) Xt + θ)ρdθ) = Xt + θ)μdθ) dt + σ t) dwt) 31) with the initial value Xt)=ξt), t [,]. Theorem 4 There is a unique strong solution {Xt), t } of equation 31), and the solution can be reresented exlicitly by Xt)=Zt)ξ) + Zt + θ)ξ)ρdθ)+ Zt s + θ)ξ s) dsρdθ)+ θ θ Zt + θ s)ξs) dsμdθ) Zt s)σ s) dws), 3) in which {Zt), t } is the fundamental solution of equation 8) with the initial value Z) = 1 and Zθ)=for θ [,]. The following two theorems rovide the th moment exonential stability and almost sure exonential stability of the solutions to 31). Theorem 5 Assume that ῡ <and the following condition holds: e γ s σ s) ds < for some γ >. 33) Then there exist ositive constants C and κ such that, for each >,the solution of equation 31) satisfies E Xt) C e κt, t. 34) Proof For a i R, i =1,,...,n, a 1 + a + + a n ) a 1 + a + + a n ) n 1 a 1 + a + + a n ), 35) we have E Xt) 5 1 E Zt)ξ) + E Zt + θ)ξ)ρdθ) + E Zt s)σ s) dws) + E Zt + θ s)ξs) dsμdθ) θ ) + E Zt + θ s)ξ s) dsρdθ) θ =: 5 1 K 1 t)+k t)+k 3 t)+k 4 t)+k 5 t) ). 36)

9 Liuand SongAdvances in Difference Equations 15) 15:177 Page 9 of 1 We shall study K i t), i =1,...,5, resectively. Similarly to the roof of Theorem, there exists ᾱ >suchthat K 1 t) c ᾱ e ᾱt ξ, 37) K 3 t) / Ɣ +1 ) Ɣ 1 ) c 1 e λ t 38) and K 4 t) 1/)) c ᾱ eᾱτ ξ μ var [,] )) e ᾱt. 39) Then we just need to consider K t)andk 5 t). For K t), by Hölder s inequality and inequality 3), we have K t) E Zt + θ) ξ) ρ var dθ) ρ var )) 1 E [,] Zt + θ) ξ) ) ρ var dθ) c ᾱ eᾱτ ρ var [,] )) e ᾱt. 4) For K 5 t), by Hölder s inequality and inequality 3), we get K 5 t) E Zt + θ s) ξ s) ds μ var dθ) θ [ E Zt + θ s) ξ s) ) 1 ) 1 q ] ds) μ var dθ) μ var dθ) θ [ = E Zt + θ s) ξ s) ) ) ] μ var )) 1 ds μ var dθ) [,] θ [ E θ 1 Zt + θ s) ξ s) ] μ var )) 1 ds μ var dθ)) [,] θ c ᾱ e ᾱt )) 1 μ var [,] θ 1 e ᾱθ s) ξ s) ds μ var dθ). 41) Then, by combining 37)-41), we conclude that E Xt) 5 1 c ᾱ e ᾱt ξ + 1/)) c ᾱ eᾱτ ξ )) e μ var [,] ᾱt + c ᾱ eᾱτ ρ var [,] )) e ᾱt + / Ɣ +1 ) Ɣ 1 ) c 1 e λ t + c ᾱ e ᾱt )) 1 μ var [,] θ 1 C e κt, θ θ e ᾱθ s) ξ s) ) ds μ var dθ)

10 Liuand SongAdvances in Difference Equations 15) 15:177 Page 1 of 1 where κ := minᾱ, λ )and C =5 1 c ᾱ ξ + 1/)) c ᾱ eᾱτ ξ )) / Ɣ +1 μ var [,] + ) Ɣ 1 ) c 1 + c )) 1 ᾱ μ var [,] θ 1 e ᾱθ s) ξ s) ds μ var dθ) + c ᾱ eᾱτ ρ var [,] )) ) <. θ This comletes the roof. Theorem 6 Assume that ῡ <and that 33) is satisfied. Then the solution of 31) is almost surely exonentially stable, i.e., there exists β >such that lim su t 1 t log Xt) β, a.s. 4) Proof Note that for n 1 t n, n 1, Xt)canbereresentedas n Xt)=Xn 1)+ ) Xs + θ)ρdθ) ds + σ s) dws) n ) + Xs + θ)μdθ) ds. 43) With inequality 35), it followsthat E su t n Xt) ) 4 E Xn 1) n + E n +E Xs + θ)μdθ) +E su σ s) dws) t n ) ) ) Xs + θ)ρdθ) ds ) ds )) =: 4 L 1 t)+l t)+l 3 t)+l 4 t) ). 44) We shall study L i t)fori =1,,3,4,resectively.ForL 1 t), by inequality 34), it is obvious that L 1 t) C e κ). 45) For L t), by Hölder s inequality and inequality 34), we have n ) ) L t) E Xs + θ) ρ var dθ) ds n ) E Xs + θ) ρ var dθ)) ds { n [ ) 1 E ρ var dθ) Xs + θ) 1 ] } ρ var dθ)) ds

11 Liuand SongAdvances in Difference Equations 15) 15:177 Page 11 of 1 = ρ var [,] ) n ρ var [,] ) n E Xs + θ) ρ var dθ) ds C e κs+θ) ρ var dθ) ds )) e C ρ var [,] κτ e κ). 46) Alying similar techniques to L 3 t), we get L 3 t) C μ var [,] )) e κτ e κ). 47) For L 4 t), by the Burkholder-Davis-Gundy inequality, there exist c >andλ 3 such that L 4 t) c 3 e λ 3), 48) where c 3 := c σ s) e λ 3s ds <. Then, by combining 45)-48), we conclude that E su t n Xt) ) 4 C e κ) + C ρ var [,] )) e κτ e κ) + C μ var [,] )) e κτ e κ) + c 3 e λ 3) ) c 5 e λ 6n, 49) where λ 6 := min κ,λ 3 )and c 5 =4 C e κ + C ρ var [,] )) e κτ e κ + C μ var [,] )) e κτ e κ + c 3 e λ 3 ). Then, in order to carry out arguments analogous to those of 3)and4), for any λ 7 < λ 6, Xt) e λ 7n e λ 7t. 5) The desired conclusion 4)issatisfiedwith β = λ 7. Examle Consider the linear neutral SDE d Xt)+ 13 ) Xt 1) = Xt 1)dt + a 1 Xt + θ) dθ dwt), Xt)=ξt), 51) where a R and {Wt)} t is a real-valued Brownian motion defined on the robability sace, P, F, {F t } t ). The characteristic equation associated with the deterministic counterart of 51)is λ + 1+ λ ) e λ =, λ C. 5) 3 A calculation by the Matlab shows that the unique root of 5)isλ = Thus, from this condition, together with 8), we deduce that the solution of 31)isth moment exonentially stable and almost surely exonentially stable by Theorem 5 and Theorem 6.

12 Liuand SongAdvances in Difference Equations 15) 15:177 Page 1 of 1 Remark 1 In this aer, for notational simlicity, we only treated asymtotic behavior of samle aths for two classes of real-valued retarded SDEs without dissiativity. Our results can be readily generalized to the multidimensional cases. The key is the use of a multidimensional variation-of-constants formula. Cometing interests The authors declare that they have no cometing interests. Authors contributions All authors contributed equally to the manuscrit. All authors read and aroved the final manuscrit. Acknowledgements We are very grateful to the anonymous referees and the associate editor for their careful reading and helful comments. Received: February 15 Acceted: 19 May 15 References 1. Yorke, JA: Asymtotic stability for one dimensional differential-delay equations. J. Differ. Equ. 71), ). Mao, X: Exonential stability of equidistant Euler-Maruyama aroximations of stochastic differential delay equations. J. Comut. Al. Math. 1), ) 3. Hou, Z, Bao, J, Yuan, C: Exonential stability of energy solutions to stochastic artial differential equations with variable delays and jums. J. Math. Anal. Al. 3661), ) 4. Scheutzow, M: Exonential growth rate for a singular linear stochastic delay differential equation. arxiv: ) 5. Bao, J, Yin, G, Wang, L, Yuan, C: Exonential mixing for retarded stochastic differential equations. arxiv: ) 6. Bao, J, Yin, G, Yuan, C: Exonential ergodicity for retarded stochastic differential equations. Al. Anal. 9311), ) 7. Bao, J, Hou, Z, Yuan, C: Stability in distribution of mild solutions to stochastic artial differential equations. Proc. Am. Math. Soc. 1386), ) 8. Mao, X, Shen, Y, Yuan, C: Almost surely asymtotic stability of neutral stochastic differential delay equations with Markovian switching. Stoch. Process. Al. 1188), ) 9. Bao, J, Hou, Z, Yuan, C: Stability in distribution of neutral stochastic differential delay equations with Markovian switching. Stat. Probab. Lett. 7915), ) 1. Wu, M, Huang, NJ, Zhao, CW: Stability of half-linear neutral stochastic differential equations with delays. Bull. Aust. Math. Soc. 83), ) 11. Bao, J, Hou, Z: Existence of mild solutions to stochastic neutral artial functional differential equations with non-lischitz coefficients. Comut. Math. Al. 591), ) 1. Milosevic, M: Almost sure exonential stability of solutions to highly nonlinear neutral stochastic differential equations with time-deendent delay and the Euler-Maruyama aroximation. Math. Comut. Model. 573), ) 13. Bao, J, Yin, G, Yuan, C: Stationary distributions for retarded stochastic differential equations without dissiativity. arxiv: ) 14. Bao, J, Hou, Z, Wang, F: Exonential stability in mean square of imulsive stochastic difference equations with continuous time. Al. Math. Lett. 5), ) 15. Bao, J, Truman, A, Yuan, C: Almost sure asymtotic stability of stochastic artial differential equations with jums. SIAM J. Control Otim. 49), ) 16. Bao, J, Yuan, C: Numerical aroximation of stationary distributions for stochastic artial differential equations. J. Al. Probab. 513), ) 17. Bao, J, Yuan, C: Large deviations for neutral functional SDEs with jums. Stochastics 871), ) 18. Dung, N: Asymtotic behavior of linear fractional stochastic differential equations with time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 191), ) 19. Hale, JK: Introduction to Functional Differential Equations. Sringer, New York 1993). Reiß, M, Riedle, M, Gaans, OV: Delay differential equations driven by Lévy rocesses: stationarity and Feller roerties. Stoch. Process. Al. 1161), ) 1. Mao, X: Stochastic Differential Equations and Alications. Horwood, Chichester 7)