Guangzhou, P.R. China

Transcription

1 This article was downloaded by:[luo, Jiaowan] On: 2 November 2007 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Stochastic Analysis and Applications Publication details, including instructions for authors and subscription information: Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable Jiaowan Luo a ; Leonid Shaikhet b a School of Mathematics and Information Science, Guangzhou University, Guangzhou, P.R. China b Department of Higher Mathematics, Donetsk State University of Management, Donetsk, Ukraine Online Publication Date: 01 November 2007 To cite this Article: Luo, Jiaowan and Shaikhet, Leonid (2007) 'Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable', Stochastic Analysis and Applications, 25:6, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Stochastic Analysis and Applications, 25: , 2007 Copyright Taylor & Francis Group, LLC ISSN print/ online DOI: / Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable Jiaowan Luo School of Mathematics and Information Science, Guangzhou University, Guangzhou, P.R. China Leonid Shaikhet Department of Higher Mathematics, Donetsk State University of Management, Donetsk, Ukraine Abstract: The general method of Lyapunov functionals construction, that was proposed by Kolmanovskii and Shaikhet and successfully used already for functional-differential equations, difference equations with discrete time, difference equations with continuous time, and is used here to investigate the stability in probability of nonlinear stochastic Volterra difference equations with continuous time. It is shown that the investigation of the stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to investigation of the asymptotic mean square stability of the linear part of this equation. Keywords: Asymptotic mean square stability; Continuous variable; Method of Lyapunov functionals construction; Nonlinear stochastic Volterra difference equations; Stability in probability. Mathematics Subject Classification (2000): 39A11; 37H10. Received March 18, 2006; Accepted September 29, 2006 Jiaowan Luo has been partially supported by NNSF of China (Grant No ). Address correspondence to Jiaowan Luo, School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong , P.R. China;

3 1152 Luo and Shaikhet 1. INTRODUCTION Difference equations with continuous variable are difference equations in which the unknown function is a function of a continuous variable. (The term difference equations is usually used for difference equations with discrete variable.) In practice, time often is involved as the independent variable in difference equations with continuous variable. In view of this fact, we may also refer to them as difference equations with continuous time. Difference equations with continuous variable appear as natural descriptions of observed evolution phenomena in many branches of the natural sciences (see, e.g., [5, 12]). Deterministic and stochastic difference equations with continuous variable are enough popular with researchers [1, 4, 5, 7, 9 11]. For example, Korenevskii [4] made a first attempt to study spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the mean-square asymptotic stability of solutions to systems of linear stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For some results on the oscillation of ordinary difference equations with continuous variable, we refer to Domshlak [1]. For the asymptotic results of ordinary difference equations with continuous variable, we refer to Philos and Purnaras [7], stability conditions for linear and nonlinear stochastic Volterra difference equations with continuous time were obtained by Shaikhet [9 11]. Motivated by the results in Shaikhet [8], concerning the stability in probability of solutions of nonlinear stochastic differential equations, and in Paternoster and Shaikhet [6], concerning the stability in probability of solutions of nonlinear stochastic Volterra difference equations with discrete time, in the present article, we will be interested in the stability in probability of nonlinear stochastic Volterra difference equations with continuous variable. It is noticed that the above articles [4, 9 11] deal with the mean-square stability which is different from the stability in probability we are concerned in this article. Similar to [6, 8], here it will be shown that investigation of the stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to investigation of the asymptotic mean square stability of the linear part of this equation. The conditions of the stability in probability are obtained here via the general method of Lyapunov functionals construction, that was proposed by Kolmanovskii and Shaikhet and successfully used already for functional-differential equations, for difference equations with discrete time and for difference equations with continuous time [2, 3, 6, 8 11].

4 Stability of Stochastic Difference Equations PRELIMINARIES AND GENERAL THEOREMS Let F P be a probability space and F t t t0 be a nondecreasing family of sub--algebras of F, i.e., F t1 F t2 for t 1 <t 2, E be the mathematical expectation with respect to P, P t = P /F t and E t = E /F t be the conditional probability and the conditional expectation. Consider a stochastic Volterra difference equation with unbounded delay xt + = a i xt h i + b i xt h i t + with the initial condition + gt xt xt h 1 xt h t > t 0 (2.1) x = = t 0 h m t 0 (2.2) Here [ ] t t0 = m + (2.3) m is a given nonnegative integer, t denotes the largest integer less than or equal to t when t 0, and t is the smallest integer greater than or equal to t when t<0. The time increment per iteration step is a positive constant, the delays h i, i = 0 1, are constants satisfying the rational relation h i = i, a i b i, i = 0 1 are known constants, the functional g satisfies the condition gt xt xt h 1 xt h i xt h i i t>t 0 i 0 i > 1 i = 0 1 (2.4), isaf t0 -measurable function, the perturbation t R is a F t -measurable stationary stochastic process with conditions E t t + = 0 E t 2 t + = 1 t>t 0 (2.5) A solution of problem (2.1) (2.2) is a F t -measurable process xt = xt t 0, which is equal to the initial function t from (2.2) for t t 0 and with probability 1 is defined by Equation (2.1) for t>t 0. Definition 2.1. The trivial solution of Equations (2.1) (2.2) is called stable in probability if for any >0 1 > 0 and t 0 0, there exists a = 1 t 0 >0 such that { } P t0 sup xt > < 1 (2.6) t t 0

5 1154 Luo and Shaikhet for any initial function which are less than with probability 1, that is, where 0 = sup. P { 0 < } = 1 (2.7) Definition 2.2. The trivial solution of Equations (2.1) (2.2) is called mean square stable if for any >0and t 0 0 there exists a = t 0 >0such that Ext 2 <for all t t 0 if 2 1 = sup E 2 <. If besides, lim t Ext 2 = 0 for all initial functions, then the trivial solution of Equations (2.1) (2.2) is called asymptotically mean square stable. Theorem 2.1. Let there exist a nonnegative functional Vt = Vt xt xt h 1 xt h and positive numbers c 1 c 2, such that t > t 0 Vt c 1 x 2 t t t 0 (2.8) Vs c 2 sup x 2 s t 0 t 0 (2.9) s and E t Vt 0 t>t 0 (2.10) if xs U = x x s t where Vt = Vt + Vt (2.11) Then the trivial solution of Equations (2.1) (2.2) is stable in probability. Proof. We will show that for any positive numbers and 1, there exists a positive number such that the solution of Equation (2.1) satisfies (2.6) with the initial condition (2.2) satisfies condition (2.7). Let xt, t t 0, be a solution of Equation (2.1). Consider the random variable T such that T = inft t 0 xt > (2.12) and two events: sup t t0 xt >and xt. It is easy to see that { } sup xt > xt (2.13) t t 0

6 Stability of Stochastic Difference Equations 1155 Put [ T t0 q = [ T t0 ] From (2.10) we have q E T i VT i = E t0 ] + 1 if [ ] T t0 if [ ] T t0 = T t 0 < T t 0 q E t0 VT i 1 VT i (2.14) = E t0 VT Vs 0 (2.15) where s = T q t 0 t 0. So, using (2.13), Chebyshev inequality, (2.7) (2.9), and (2.15), we get { } P t0 sup xt > P t0 xt > E t 0 x 2 T E t 0 VT t t 0 2 c 1 2 Vs c 1 c c 2 2 (2.16) 2 c 1 2 c 1 2 Choosing such that = 1 c 1 /c 2, we get (2.6). Remark 2.1. It is easy to see that if 0, then Psup t t0 xt > Psup t t0 xt > 0. It means that if condition (2.6) holds for enough small >0, then it holds for every >0. Thus, for the stability in probability of the trivial solution of Equations (2.1) (2.2), it is sufficient to prove condition (2.6) for enough small >0. From Theorem 2.1, it follows that an investigation of the stability in probability of the trivial solution of Equation (2.1) can be reduced to construction of appropriate Lyapunov functionals. 3. CONDITIONS OF STABILITY IN PROBABILITY Denote Theorem 3.1. a = a i b = b i = i (3.1) Let < and a 2 + b 2 < 1 (3.2) Then the trivial solution of Equations (2.1) (2.2) is stable in probability.

7 1156 Luo and Shaikhet Proof. Using the general method of Lyapunov functionals construction for Equations (2.1) (2.2) we will construct Lyapunov functional in the form Vt = V 1 t + V 2 t, where V 1 t = x 2 t. Via (2.1), (2.5) for t>t 0 we have E t V 1 t = E t x 2 t + x 2 t = E t (gt xt xt h 1 xt h = + a i xt h i + ( 2 a i xt h i ) + b i xt h i t + ) 2 x 2 t ( b i xt h i + g 2 t xt xt h 1 xt h + 2 a i xt h i gt xt xt h 1 xt h x 2 t ) 2 (3.3) Using (3.1) we obtain Via (2.4) we have and ( ( 2 a i xt h i ) a 2 b i xt h i ) b a i x 2 t h i (3.4) b i x 2 t h i (3.5) g 2 t xt xt h 1 xt h i xt h i 2 i 2 i 2i 1 x 2 t h i (3.6) a i xt h i gt xt xt h 1 xt h 2 a i xt h i k xt h k k k=0

8 Stability of Stochastic Difference Equations 1157 = a i k xt h k k 1( 2xt hk xt h i ) k=0 a i k xt h k k 1( xt hk 2 +xt h i 2) k=0 = a i k xt h k k 1 xt hk 2 + k=0 a i k=0 Assume that xs U for s t and put k = If 1 then k <. So, k xt h k k 1 xt hi 2 (3.7) i ki 1 k = 1 2 (3.8) where a i xt h i gt xt xt h 1 xt h 2 a k k 1 xt h k a i xt h i 2 k=0 a i i a i x 2 t h i (3.9) As a result we obtain E t V 1 t A 0 1x 2 t + A i x 2 t h i (3.10) A i = a + 1 a i +bb i +a + i 1 i i 1 i = 0 1 (3.11) Choosing now the functional V 2 in the form ( ) V 2 t = x 2 t h i A l (3.12) l=i

9 1158 Luo and Shaikhet we have V 2 t = +1 ( ) ( ) x 2 t + h i A l x 2 t h i A l ( = x 2 t h i l=i+1 l=i l=i ) ( ) A l x 2 t h i A l = x 2 t A i A i x 2 t h i (3.13) From (3.10), (3.11), and (3.13) for Vt = V 1 t + V 2 t we have E t Vt a 2 + b 2 + 2a x 2 t t > t 0 (3.14) From (3.2), it follows that E t Vt 0 for enough small. It is easy to see that the functional Vt satisfies the conditions of Theorem 2.1. Therefore, using Remark 2.1, we get that the trivial solution of Equations (2.1) (2.2) is stable in probability. l=i Let us obtain now another condition of the stability in probability of the trivial solution of Equations (2.1) (2.2). Denote = a j = a j (3.15) j=i j=0 Theorem 3.2. Let < and b 2 < 1 (3.16) Then the trivial solution of Equations (2.1) (2.2) is stable in probability. Proof. form It is easy to see that Equation (2.1) can be represented in the xt + = xt + Ft + b i xt h i t + + gt xt xt h 1 xt h (3.17) where Ft = xt h l a i Ft = Ft + Ft (3.18) i=l

10 Stability of Stochastic Difference Equations 1159 Following the general method of Lyapunov functionals construction we will construct Lyapunov functional Vt in the form Vt = V 1 t + V 2 t again but now with Calculating E t V 1 t we get V 1 t = xt Ft 2 E t V 1 t = E t xt + Ft + 2 xt Ft 2 = E t (xt Ft + b i xt h i t + ) 2 + gt xt xt h 1 xt h xt Ft 2 ( 2 = 2 1x 2 t + b i xt h i ) + 21 xtft + g 2 t xt xt h 1 xt h + 2xtgt xt xt h 1 xt h 2Ftgt xt xt h 1 xt h (3.19) Using the assumption xs U = { xx } for s t, inequalities (3.5), (3.6), and 2xtFt a i x 2 t + x 2 t h l x 2 t + a i x 2 t h l i=l 2xtgt xt xt h 1 xt h i xt h i i 1 x 2 t + x 2 t h i x 2 t + l l 1 x 2 t h l 2Ftgt xt xt h 1 xt h i a j i 1 x 2 t h l + x 2 t h i j=l ( ) x 2 t + 1 a j + l l 1 x 2 t h l j=l i=l

11 1160 Luo and Shaikhet we have E t V 1 t ( bb ) 0 1 x 2 t + A i x 2 t h i (3.20) with A i = bb i + i 2i a j + +i i 1 j=i (3.21) Define now the functional V 2 by (3.12), (3.21). Then using (3.20), (3.13), similar to (3.14) we get E t Vt ( b ) x 2 t (3.22) Here t>t 0, 1 and 2 are defined in (3.8). From here and (3.16) it follows that for enough small there exists c 3 > 0 such that E t Vt c 3 x 2 t t > t 0 (3.23) From (3.23) it follows that the functional Vt satisfies condition (2.10). It is easy to check that the functional Vt also satisfies condition (2.9); but it does not satisfy condition (2.8). So we cannot here use Theorem 2.1 and have to find another way to prove. Consider the random variable T which is described by (2.12). From (2.16) we have { } P t0 sup xt > P t0 xt E t 0 x 2 T (3.24) t t 0 2 To estimate E t0 x 2 T note that VT xt FT 2 x 2 T 2xTFT (3.25) Besides via (3.18), (3.15), and (2.14) we have 2xTFT mt a j x 2 T + x 2 T h l j=l q 1 x 2 T + a j x 2 T h l + j=l mt l=q a j x 2 T h l j=l

12 Stability of Stochastic Difference Equations 1161 Since T h l t 0 for l q then via (2.7) we obtain q 1 2xTFT x 2 T + a j x 2 T h l + 2 (3.26) From (3.16) it follows that <1. So, substituting (3.26) into (3.25) we obtain q 1 E t0 x 2 T 1 {E 1 t0 VT + a j E t0 x 2 T h l + } 2 (3.27) Note that t 0 <T h l <T for 1 l<q. So, from (3.23) it follows c 3 E t0 x 2 T h l E t0 VT h l. Besides, similar to (2.15) one can show that E t0 VT h l Vs, where 0 l<q, s = T q t 0. Hence, via (3.27), (2.9), (3.15), (2.7) we have E t0 x 2 T 1 1 {Vs + j=l j=l { q 1 1 c } q Vs a j + 2 j=l c 3 { 1 1 c 2 + c } 2 + c 3 j=l a j c c } 2 = C 2 (3.28) From here and (3.24) by = 1 /C, (2.6) follows. Therefore, the trivial solution of Equations (2.1) (2.2) is stable in probability. Remark 3.1. Note that condition 3.16 can be transformed to the form b 2 < < 1 Remark 3.2. As was shown in [8] for stochastic differential equations with delays and in [6] for stochastic difference equations with discrete time one can show that inequalities (3.2) and (3.16) are sufficient conditions for the asymptotic mean square stability of the trivial solution of Equations (2.1) (2.2) in the case when g 0. It means that investigation of the stability in probability of nonlinear stochastic difference equations with continuous variable with order of nonlinearity more than one can be reduced to the investigation of the asymptotic mean square stability of the linear part of this equation.

13 1162 Luo and Shaikhet 4. EXAMPLES Example 4.1. Consider the nonlinear difference equation xt + = 2 i+1[ 1 i a + bt + ] xt h i + 2 i+1 x i t h i (4.1) where = m + t t 0, m 0, h i = i, i > 1, i = 0 1. From Theorem 3.1 it follows that if a 2 + b 2 < 1 (4.2) then the trivial solution of Equation (4.1) is stable in probability. To use Theorem 3.2 note that = 3 1 a, = 3 1 a. Via Remark 3.1 we get condition of the stability in probability in the form { a if a 0 3 b < a1 + a if a 1 0 (4.3) On Figure 1 the stability regions are shown obtained by condition (4.2) (Curve No. 1) and condition (4.3) (Curve No. 2). One can see that both these regions supplement each other. Figure 1. Stability regions for Equation (4.1).

14 Stability of Stochastic Difference Equations 1163 Example 4.2. Consider the nonlinear difference equation xt + = axt + bxt kt + (4.4) 1 + sinxt where = m + t t 0, m 0, k 0 and For the functional Vt = x 2 t + b 2 < 1 (4.5) k x 2 t j j=1 via inequalities (4.5) and sinx 1 we have k k E t Vt = E t x 2 t + + b 2 x 2 t + j x 2 t b 2 x 2 t j j=1 ( ) axt 2 = E t + bxt kt sinxt x 2 t + b 2 x 2 t x 2 t k a 2 x 2 t = 1 + sinxt 2 x2 t + b 2 x 2 t ( ) a b2 1 x 2 t j=1 From [10] it follows that the inequality a b2 < 1 (4.6) is a sufficient condition for the asymptotic mean square stability of the trivial solution of Equation (4.4). To get a sufficient condition for the stability in probability of the trivial solution of Equation (4.4) rewrite it in the form where xt + = axt + gxt xt + bxt k i (4.7) ax siny gx y = 1 + siny

15 1164 Luo and Shaikhet Using the inequality siny y in numerator of the fraction and conditions (4.5), siny 1 in the denominator we have gx y axy 1 a 21 x2 + y 2 It means that the function gx y satisfies condition (2.4). Therefore, the condition a 2 + b 2 < 1 which can be obtained from (4.6) by = 0, is a sufficient condition for the asymptotic mean square stability of the linear part of Equation (4.7) and as it follows from Theorem 2.1 is a sufficient condition for the stability in probability of the trivial solution of Equation (4.4) for all satisfying condition (4.5). REFERENCES 1. Domshlak, Y Oscillatory properties of linear difference equations with continuous time. Differential Equations Dynamic Systems 1(4): Kolmanovskii, V.B., and Shaikhet, L.E General method of Lyapunov functionals construction for stability investigations of stochastic difference equations. In Dynamical Systems and Applications. World Scientific Series in Applicable Analysis 4. World Sci. Publ., River Edge, NJ, pp Kolmanovskii, V.B., and Shaikhet, L.E A method of Lyapunov functionals construction for stochastic differential equations of neutral type. Differential Equations 31(11): Korenevskii, D.G Stability criteria for solutions of systems of linear deterministic or stochastic delay difference equations with continuous time. Mathematical Notes 70(2): Maistrenko, Yu.L., and Sharkovsky, A.N Difference equations with continuous time as mathematical models of the structure emergences. Dynamical Systems and Environmental Models. Eisenach, Mathem. Ecol., Akademie-Verlag, Berlin, pp Paternoster, B., and Shaikhet, L About stability of nonlinear stochastic difference equations. Applied Mathematics Letters 13: Philos, Ch.G., and Purnaras, I.K An asymptotic results for some delay difference equations with continuous variable. Advances in Difference Equations 2004(1): Shaikhet, L Stability in probability of nonlinear stochastic hereditary systems. Dynamic Systems and Applications 4(2): Shaikhet, L Lyapunov functionals construction for stochastic difference second kind Volterra equations with continuous time. Advances in Difference Equations 2004(1):67 91.

16 Stability of Stochastic Difference Equations Shaikhet, L Construction of Lyapunov functionals for stochastic difference equations with continuous time. Mathematics and Computers in Simulation 66(6): Shaikhet, L General method of Lyapunov functionals construction in stability investigations of nonlinear stochastic difference equations with continuous time. Stochastics and Dynamics 5(2): Sharkovsky, A.N., Maistrenko, Yu.L., and Romanenko, E.Yu Difference Equations and Their Applications, Mathematics and Its Applications, vol Kluwer Academic, Dordrecht.