Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 132923, 7 pages http://dx.doi.org/10.1155/2013/132923
Research Article Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations Junfei Cao,1 Zaitang Huang,2 and Caibin Zeng3 1
Department of Mathematics, Guangdong University of Education, Guangzhou 510310, China School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530023, China 3 Department of Mathematics, South China University of Technology, Guangzhou 510640, China 2
Correspondence should be addressed to Junfei Cao;
[email protected] Received 2 August 2013; Accepted 21 August 2013 Academic Editors: K. Ammari and S. Kim Copyright Š 2013 Junfei Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A class of stochastic differential equations given by đđĽ(đĄ) = đ(đĽ(đĄ))đđĄ+đ(đĽ(đĄ))đđ(đĄ), đĽ(đĄ0 ) = đĽ0 , đĄ0 ⤠đĄ ⤠đ < +â, are investigated. Upon making some suitable assumptions, the existence and uniqueness of solution for the equations are obtained. Moreover, the existence and uniqueness of solution for stochastic Lorenz system, which is illustrated by example, are in good agreement with the theoretical analysis.
1. Introduction Stochastic differential equations (SDEs) play an important role in characterizing many social, physical, biological, and engineering problems. They are important from the viewpoint of applications since they incorporate (natural) randomness into the mathematical description of the phenomena and provide a more accurate description of it. Therefore, the theory of SDEs has developed quickly, the investigation for SDEs has attracted considerable attention of researchers, and many qualitative theories of SDEs have been obtained (see [1â9] and the references therein). The existence and uniqueness of solution are among the most basic and key topics in qualitative theory of SDEs. In the last two decades, the existence and uniqueness of solution for SDEs have been considered in many publications such as [10â14] and the references therein. Especially, Mao had investigated the stochastic differential equations: đđ (đĄ) = đ (đ (đĄ) , đĄ) đđĄ + đ (đ (đĄ) , đĄ) đđľđĄ ,
đ (đĄ0 ) = đ0 , (1)
on the closed interval [đĄ0 , đ], đĄ0 ⤠đ, in his book [14], and obtained that if Lipschitz condition
óľŠ2 óľŠ óľŠ2 óľŠóľŠ óľŠóľŠđ (đ, đĄ) â đ (đ, đĄ)óľŠóľŠóľŠ ⨠óľŠóľŠóľŠđ (đ, đĄ) â đ (đ, đĄ)óľŠóľŠóľŠ óľŠ óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠ2 ⤠đžóľŠóľŠóľŠđ â đóľŠóľŠóľŠ , đ, đ â Rđ , đĄ â [đĄ0 , đ] ,
(2)
and linear growth condition: óľŠ2 óľŠ óľŠ2 óľŠóľŠ 2 óľŠóľŠđ (đ, đĄ)óľŠóľŠóľŠ ⨠óľŠóľŠóľŠđ (đ, đĄ)óľŠóľŠóľŠ ⤠đž (1 + âđâ ) , (đ, đĄ) â Rđ Ă [đĄ0 , đ] ,
(3)
hold, then (1) had a unique solution đ(đĄ) satisfying đ(đĄ) â M2 ([đĄ0 , đ], Rđ ). Furthermore, Mao [14] also discussed stochastic functional differential equations with finite delay: đđ (đĄ) = đ (đđĄ , đĄ) đđĄ + đ (đđĄ , đĄ) đđľđĄ , đđĄ0 = đ,
đĄ0 ⤠đĄ ⤠đ < +â,
(4)
where đđĄ = {đ(đĄ + đ) : âđ ⤠đ ⤠0} could be considered as a đś([đĄ0 , đ], Rđ )-value stochastic process. The initial value đđĄ0 = đ = {đ(đ) : âđ ⤠đ ⤠0} is an đšđĄ0 -measurable
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đś([đĄ0 , đ], Rđ )-value random variable such that Eâđâ2 < â. For (4), if uniform Lipschitz condition óľŠ2 óľŠ2 óľŠ óľŠóľŠ óľŠóľŠđ (đ, đĄ) â đ (đ, đĄ)óľŠóľŠóľŠ ⨠óľŠóľŠóľŠđ (đ, đĄ) â đ (đ, đĄ)óľŠóľŠóľŠ óľŠ2 óľŠ â¤ đžóľŠóľŠóľŠđ â đóľŠóľŠóľŠ , đž > 0,
(5)
for any đ, đ â đś([đĄ0 , đ]; Rđ ), đĄ â [đĄ0 , đ], and linear growth condition óľŠ2 óľŠ óľŠ2 óľŠóľŠ óľŠ óľŠ2 óľŠóľŠđ (đ, đĄ)óľŠóľŠóľŠ ⨠óľŠóľŠóľŠđ (đ, đĄ)óľŠóľŠóľŠ ⤠đž (1 + óľŠóľŠóľŠđóľŠóľŠóľŠ ) , (6) (đ, đĄ) â Rđ Ă [đĄ0 , đ] , đž > 0, are satisfied, then (4) had a unique solution đ(đĄ); moreover, đ(đĄ) â M2 ([đĄ0 , đ], Rđ ). We also mention that, following Mao [14], many papers were devoted to improving the results of Mao [14] by weakening the Lipschitz conditions on coefficients (e.g., see Jiang and Wang [15], Fei [16], Caraballo et al. [17], Wu et al. [18], Jiang and Shen [19], Taniguchi [20], Fan and Jiang [21], Bao and Hou [22], Ren et al. [23], Taniguchi [24], Wang and Huang [25], Lin [26], Xie [27], Fei [28], Ren and Zhang [29] and Zhang [30], etc.). In particular, Caraballo et al. [17], Taniguchi [20], Bao and Hou [22], and Taniguchi [24] have studied the existence and uniqueness of solutions to SDEs under the non-Lipschitzian condition. To the best of our knowledge, most of the results on existence theory for SDEs focused on the case of the coefficient satisfying linear growth condition; however, the results on existence theory for SDEs without linear growth condition were discussed seldom. In this paper, without linear growth condition, some new criteria ensuring the existence and uniqueness of solutions for a class of SDEs are firstly established. These criteria improve, complement a number of existing results, and handle some cases not covered by known criteria. The results obtained in this paper not only improve the previous conclusion in the case of linear growth condition, but also can be applied in the existence and uniqueness of solution for the stochastic Lorenz system for weather forecasting, some other systems, and so on. The rest of this paper is organized as follows. In Section 2, some relating notations and preliminary facts are introduced. Section 3 obtains the existence and uniqueness of solution for a class of SDEs without linear growth condition. In Section 4, an interesting examples is given to show the effectiveness of our results.
of the đ-field generated by {đ(đĄ) â đ(đĄ0 ) : đĄ0 ⤠đĄ ⤠đ} and contains all P-null sets. đ(đĄ) is a given đ-dimensional standard Brownian motion. Consider đ-dimensional stochastic differential equations: đđĽ (đĄ) = đ (đĽ (đĄ)) đđĄ + đ (đĽ (đĄ)) đđ (đĄ) , đĽ (đĄ0 ) = đĽ0 ,
đĄ0 ⤠đĄ ⤠đ < +â,
(7)
where đĽ(đĄ) = (đĽ1 (đĄ), đĽ2 (đĄ), . . . , đĽđ (đĄ))⤠, đĽ0 = (đĽ1 (đĄ0 ), đĽ2 (đĄ0 ), . . . , đĽđ (đĄ0 ))⤠is independent of FđĄ , and satisfies EâđĽ0 â2 < +â, đ(đĄ) is an đ-dimensional Wiener process, đ(đĽ(đĄ)) = (đ1 (đĽ(đĄ)), đ2 (đĽ(đĄ)), . . . đđ (đĽ(đĄ)))⤠, đ(đĽ(đĄ)) is a đ Ă đ matrix. Our purpose is to find the solution for (7). Hence, we will show the existence, uniqueness theorem and the properties of solution for (7) in the next section. Moreover, to illustrate the effectiveness of our results, we prove the existence and uniqueness of solution for the stochastic Lorenz system. To guarantee the existence and uniqueness of solution for (7), the following conditions, instead of Lipschitz and linear growth conditions, are described. (đť1 ) đ(đĽ(đĄ)) satisfies the Lipschitz condition; moreover, â¨đ (đĽ) , đĽâŠ ⤠đ1 + đ2 âđĽâ2 ;
(8)
(đť2 ) đ(đĽ(đĄ)) satisfies the Lipschitz condition and the linear growth condition: óľŠ2 óľŠóľŠ 2 2 óľŠóľŠđ (đĽ)óľŠóľŠóľŠ ⤠đ (1 + âđĽâ ) ,
(9)
where â¨â
, â
⊠denotes the inner product of Rđ and đ1 , đ2 , and đ are constants.
3. The Existence and Uniqueness Theorem In this section, we start to study the existence and uniqueness of solution for (7) without linear growth condition. To complete our main results, we need to prepare several lemmas which will be utilized in the sequel. Lemma 1. Let (đť1 ) and (đť2 ) be satisfied. Setting đđ â đś1 (Rđ , R) with 1, đđ (đĽ) = { 0,
đđđ âđĽâ ⤠đ, đđđ âđĽâ ⼠đ + 1,
(10)
and đđ(đĽ) = đđ(đĽ)đ(đĽ), and then the modified stochastic differential equations:
2. Preliminaries
đđĽđ = đđ (đĽđ) đđĄ + đ (đĽđ) đđđĄ ,
This section is concerned with some notations and preliminary results which are used in what follows. In this paper, we adopt the symbols as follow: Rđ denotes the usual đ-dimensional Euclidean space, â â
â denotes norm in Rđ . If đ´ is a vector or a matrix, its transpose is denoted by đ´đ ; if đ´ is a matrix, its trace norm is represented by â â
â = âtrace(đ´đ đ´). Let (Ί, F, P) be a complete probability space with a filtration {FđĄ }đĄâ[đĄ0 ,+â) satisfying the usual conditions (i.e., it is increasing and right continuous). FđĄ0 is independent
đĽđ (đĄ0 ) = đĽ0 ,
đĄ0 ⤠đĄ ⤠đ < +â,
(11)
possesses a continuous almost sure unique FđĄ measurable solution process. Proof. As the truncation function đđ â đś1 (Rđ , R), đđ(đĽ) remains differentiable and its derivative is both continuous and has a compact support. Hence đđ(đĽ) is bounded and satisfies linear growth condition as well as Lipschitz condition.
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According to (đť2 ), đ(đĽ) satisfies linear growth condition and Lipschitz condition. Therefore, the assertion follows by the usual existence and uniqueness theorem (see Gihman and Skorohod [31], p. 37â47).
Form (đť2 ), we get the following inequality: óľŠ óľŠđ óľŠ óľŠđ EóľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ⤠EóľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ đĄ
+ ⍠[(đđ1 +
Lemma 2 (see [32]). Let đś, đˇ be both positive and semidefinite Hermite matrix, and đ is a positive integer; then đ
đ
đ
trace (đśđˇ) ⤠( trace (đś)) ( trace (đˇ)) .
đĄ0
óľŠ óľŠđâ2 Ă EóľŠóľŠóľŠđĽđ (đ ⧠đđ)óľŠóľŠóľŠ
(12)
+
0
which implies (14)
Together with (đť1 ), the differential of the đth norm is given by
(15)
where đ(đĄ) ⤠0 is an adapted process that compensates all the estimation made and could be computed explicitly. Lemma 3. Let đ â N be even, and then there exists a constant đśđ = đś(đ, đ¸âđĽ0 âđ , đ) independent of đ such that đ¸âđĽđâđ < đśđ for đĄ â [đĄ0 , đ]. Proof. For đ â N, define stopping time đđ = inf{đĄ â [đĄ0 , đ] : âđĽđ(đĄ)â ⼠đ}. Note that đĄâ§đđ
âŤ
đĄ
đ (đ ) đđ ⤠⍠đ (đ ⧠đđ) đđ for đ (đĄ) ⼠0. đĄ0
đĄ0
(16)
Since âđĽđ(đĄ ⧠đđ)âđ ⼠đđ for all đ > 0, using (đť2 ) and the above inequality, one concludes that integrand of the stochastic integral is bounded; hence, đĄâ§đđ
E (âŤ
đĄ0
óľŠ óľŠ2 EóľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ óľŠ óľŠ2 ⤠EóľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ
(20)
đĄ
óľŠ óľŠ2 + ⍠(đ1 EóľŠóľŠóľŠđĽđ (đ ⧠đđ)óľŠóľŠóľŠ + đ2 ) đđ . đĄ0
óľŠ óľŠđâ2 óľŠ óľŠđ óľŠ óľŠđ đóľŠóľŠóľŠđĽđóľŠóľŠóľŠ = (đđ1 óľŠóľŠóľŠđĽđóľŠóľŠóľŠ + đđ2 óľŠóľŠóľŠđĽđóľŠóľŠóľŠ ) đđĄ đ (đ â 1) óľŠóľŠ óľŠóľŠđâ2 óľŠóľŠ óľŠ2 + óľŠóľŠđĽđóľŠóľŠ óľŠóľŠđ (đĽđ)óľŠóľŠóľŠ đđĄ 2 óľŠ óľŠđâ2 đ đ (đĽđ) đđ (đĄ) + đ (đĄ) đđĄ, + đóľŠóľŠóľŠđĽđóľŠóľŠóľŠ đĽđ
óľŠ óľŠđâ2 + đ2 EóľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ) đđ ,
(13)
According to Lemma 2, one has óľŠ óľŠ2 óľŠ óľŠ2 đ trace (đĽđ đĽđđđ (đĽđ) đ (đĽđ)) ⤠óľŠóľŠóľŠđĽđóľŠóľŠóľŠ óľŠóľŠóľŠđ (đĽđ)óľŠóľŠóľŠ .
óľŠđ óľŠ Ă EóľŠóľŠóľŠđĽđ (đ ⧠đđ)óľŠóľŠóľŠ ] đđ . Let đ1 = đđ2 + đ2 đ(đ â 1)/2, đ2 = đđ1 + đ2 đ(đ â 1)/2, then (18) becomes óľŠđ óľŠ EóľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ óľŠ óľŠđ ⤠EóľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ đĄ (19) óľŠ óľŠđ + ⍠(đ1 EóľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ đĄ
Ă đđ (đĽđ)) đđĄ đ óľŠóľŠ óľŠóľŠđâ2 đ óľŠđĽ óľŠ trace (đ (đĽđ) đ (đĽđ)) đđĄ 2 óľŠ đóľŠ óľŠ óľŠđâ2 đ đ (đĽđ) đđ (đĄ) . + đóľŠóľŠóľŠđĽđóľŠóľŠóľŠ đĽđ
(18)
đ2 đ (đ â 1) + (đđ2 + ) 2
We compute the đth norm of the solution for (11) as follows. By ItĚo formula w.r.t. the function â(đĽ) = âđĽâđ = đ/2 (âđđ=1 đĽđ2 ) for đ being even, one obtains óľŠ óľŠđ óľŠ óľŠđâ2 đóľŠóľŠóľŠđĽđóľŠóľŠóľŠ = đóľŠóľŠóľŠđĽđóľŠóľŠóľŠ (đ (đĽđ) , đĽđ) đđĄ đ óľŠ óľŠđâ4 đ đ (đĽđ) + đ ( â 1) óľŠóľŠóľŠđĽđóľŠóľŠóľŠ trace (đĽđđĽđ 2
đ2 đ (đ â 1) ) 2
óľŠđâ2 óľŠ đóľŠóľŠóľŠđĽđ (đ )óľŠóľŠóľŠ đĽđ(đ )đ đ (đĽđ (đ )) đđ (đ )) = 0. (17)
Gronwall inequality implies that there exists a constant đś2 such that óľŠ2 óľŠ sup EóľŠóľŠóľŠđĽđ (đ ⧠đđ)óľŠóľŠóľŠ ⤠đś2 . (21) đĄâ[đĄ0 ,đ] By inductive method, there exist a constant đśđ such that óľŠ óľŠđ óľŠ óľŠđ EóľŠóľŠóľŠđĽđ (đ ⧠đđ)óľŠóľŠóľŠ ⤠EóľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ đĄ
óľ¨ óľ¨ óľŠ óľŠđ + ⍠( óľ¨óľ¨óľ¨đ1 óľ¨óľ¨óľ¨ EóľŠóľŠóľŠđĽđ (đ ⧠đđ)óľŠóľŠóľŠ đĄ
(22)
0
óľ¨ óľ¨ + óľ¨óľ¨óľ¨đ2 óľ¨óľ¨óľ¨ đśđâ2 ) đđ ⤠đśđ . It remains to show that the stopping time satisfies đđ â đ for đ â +â. By the continuity of the solution đĽđ(đĄ) in đĄ, the norm âđĽđ(đĄ ⧠đđ)âđ is bounded; therefore, it converges đ-wise to âđĽđ(đĄ)âđ as đ â +â. Since the norm is nonnegative and bounded for all đ â N, for đĄ ⤠đ, Fatou Lemma implies that óľŠđ óľŠ óľŠđ óľŠ EóľŠóľŠóľŠđĽđ (đĄ)óľŠóľŠóľŠ ⤠E ( lim óľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ) đ â +â óľŠ óľŠđ ⤠lim inf óľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ⤠đśđ . đ â +â
(23)
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Lemma 4. Let đ â N be even, and then there exists a Ěđ = đś(đ, Ě đ¸âđĽ0 âđ , đ) independent of đ such that constant đś Ěđ for đĄ â [đĄ0 , đ]. EsupđĄâ[đĄ0 ,đ] âđĽđ(đĄ)âđ < đś
(đâ2)/2
+4
óľ¨óľ¨ đĄâ§đđ óľ¨óľ¨óľ¨đ/2 óľ¨óľ¨ đ óľ¨ E sup óľ¨óľ¨âŤ 2đĽđ (đ ) đ (đĽđ (đ )) đđ(đ )óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ đĄâ[đĄ0 ,đ]óľ¨óľ¨ đĄ0
óľŠ óľŠđ = 4(đâ2)/2 đ¸óľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ + đ1 + đ2 + đ3 . (27)
Proof. From (15), it follows that óľŠ2 óľŠ óľŠ2 óľŠóľŠ óľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ = óľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ đĄâ§đđ
+âŤ
đĄ0 đĄâ§đđ
+âŤ
đĄ0 đĄâ§đM
+âŤ
đĄ0 đĄâ§đđ
+âŤ
đĄ0
óľŠ2 óľŠ 2đ2 óľŠóľŠóľŠđĽđ (đ )óľŠóľŠóľŠ đđ óľŠ óľŠ2 (2đ1 + óľŠóľŠóľŠđ (đĽđ (đ ))óľŠóľŠóľŠ ) đđ
(24)
óľ¨óľ¨ đ óľ¨đ/2 óľ¨ óľŠ2 óľ¨óľ¨ óľ¨ óľ¨óľŠ Ě1 , đ1 = 4(đâ2)/2 Eóľ¨óľ¨óľ¨óľ¨âŤ 2 óľ¨óľ¨óľ¨đ2 óľ¨óľ¨óľ¨ óľŠóľŠóľŠđĽđ (đ )óľŠóľŠóľŠ đđ óľ¨óľ¨óľ¨óľ¨ ⤠đś đ óľ¨óľ¨ đĄ0 óľ¨óľ¨ óľ¨óľ¨đ/2 óľ¨óľ¨óľ¨ đ óľ¨ óľ¨ óľŠ óľ¨ óľŠ2 đ2 = 4(đâ2)/2 Eóľ¨óľ¨óľ¨óľ¨âŤ (2 óľ¨óľ¨óľ¨đ1 óľ¨óľ¨óľ¨ + óľŠóľŠóľŠđ (đĽđ (đ ))óľŠóľŠóľŠ ) đđ óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ đĄ0 óľ¨óľ¨
đ 2đĽđ (đ ) đ (đĽđ (đ )) đđ (đ )
đ (đ ) đđ ,
which implies
+âŤ
đĄ0
+âŤ
đĄâ§đđ
đĄ0
+âŤ
đĄâ§đđ
đĄ0
đ
óľ¨ óľ¨đ/2 à ⍠(2óľ¨óľ¨óľ¨đ1 óľ¨óľ¨óľ¨ + đđ + đđ đś2 ) đđ , đĄ
óľŠ2 óľ¨ óľ¨óľŠ 2 óľ¨óľ¨óľ¨đ2 óľ¨óľ¨óľ¨ óľŠóľŠóľŠđĽđ (đ )óľŠóľŠóľŠ đđ
0
(25)
óľ¨ óľ¨ óľŠ óľŠ2 (2 óľ¨óľ¨óľ¨đ1 óľ¨óľ¨óľ¨ + óľŠóľŠóľŠđ (đĽđ (đ ))óľŠóľŠóľŠ ) đđ đ 2đĽđ (đ ) đ (đĽđ (đ )) đW (đ ) .
Taking to the power đ/2 to obtain an expression for âđĽđ(đĄ ⧠đđ)âđ , one has óľ¨ đĄâ§đđ óľŠ2 óľŠóľŠ óľŠđ óľ¨óľ¨óľŠ óľŠ2 óľ¨ óľ¨óľŠ 2 óľ¨óľ¨óľ¨đ2 óľ¨óľ¨óľ¨ óľŠóľŠóľŠđĽđ (đ )óľŠóľŠóľŠ đđ óľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ⤠óľ¨óľ¨óľ¨óľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ + ⍠óľ¨óľ¨ 0 +âŤ
đĄâ§đđ
đĄâ§đđ
đĄ0
According to obtains
đ | âđđ=1 đđ |
⤠đ
óľ¨óľ¨
âđđ=1
đ
(26)
|đđ | for đ ⼠1, one
óľŠ óľŠđ E sup óľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ đĄâ[đĄ0 ,đ] óľŠ óľŠđ ⤠4(đâ2)/2 EóľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ óľ¨óľ¨ đ óľ¨đ/2 óľ¨ óľŠ2 óľ¨óľ¨ óľ¨ óľ¨óľŠ + 4(đâ2)/2 E óľ¨óľ¨óľ¨óľ¨âŤ 2 óľ¨óľ¨óľ¨đ2 óľ¨óľ¨óľ¨ óľŠóľŠóľŠđĽđ (đ )óľŠóľŠóľŠ đđ óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ đĄ0 óľ¨óľ¨ óľ¨óľ¨ đ óľ¨óľ¨đ/2 óľ¨ óľ¨ óľŠ2 óľ¨ óľ¨ óľŠ + 4(đâ2)/2 Eóľ¨óľ¨óľ¨óľ¨âŤ (2 óľ¨óľ¨óľ¨đ1 óľ¨óľ¨óľ¨ + óľŠóľŠóľŠđ (đĽđ (đ ))óľŠóľŠóľŠ ) đđ óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ đĄ0 óľ¨óľ¨
óľ¨óľ¨đ/2 óľ¨óľ¨ đĄ óľ¨ óľ¨ đ đ3 = E sup óľ¨óľ¨óľ¨óľ¨âŤ đĽđ (đ ) đ (đĽđ (đ )) đđ (đ )óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨ đĄ đĄâ[đĄ0 ,đ]óľ¨ 0 óľ¨óľ¨ đ óľ¨đ/4 óľ¨ óľŠ óľŠ2 óľ¨óľ¨ ⤠đđ Eóľ¨óľ¨óľ¨óľ¨âŤ óľŠóľŠóľŠđĽđ (đ ) đ (đĽđ (đ ))óľŠóľŠóľŠ óľ¨óľ¨óľ¨óľ¨ đđ , óľ¨óľ¨ đĄ0 óľ¨óľ¨ đ/4
óľ¨óľ¨đ/2 óľ¨ đ 2đĽđ (đ ) đ (đĽđ (đ )) đđ (đ )óľ¨óľ¨óľ¨óľ¨ .
đâ1
Ě2 . To further estimate the which is bounded by a constant đś đ stochastic integral đ3 , using the Burkholder-Davis-Gundy (see Mao [14], p. 7), one obtains
(29)
with the constant
óľŠ2 óľ¨ óľ¨ óľŠ (2 óľ¨óľ¨óľ¨đ1 óľ¨óľ¨óľ¨ + óľŠóľŠóľŠđ (đĽđ (đ ))óľŠóľŠóľŠ ) đđ
đĄ0
+âŤ
(28)
⤠4(đâ2)/2 đ(đâ2)/2
óľŠóľŠ óľŠ2 óľŠ óľŠ2 óľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ⤠óľŠóľŠóľŠđĽ (đĄ0 )óľŠóľŠóľŠ đĄâ§đđ
Since the solution đĽđ(đĄ) for (11) is FđĄ measurable for đĄ ⼠đĄ0 and continuous in đĄ, the norm âđĽđ(đĄ)â is FđĄ â B([đĄ0 , đĄ])measurable (see Wentzell [33], p. 89, p. 18). Therefore, applying (đť2 ), together with H¨older inequality to remove the powers outside the deterministic integrals đ1 and đ2 , changing expectation and integration by Fubini theorem and using Lemma 3, one obtains
64 { { ( ) , { { { đ đ/4 đđ = { { đđ/2+1 { { ( ) , { đ/2+2 (đ/2 â 1)đ/2â1 2 {
for 0 < đ < 4, (30) for đ ⼠4.
For đ = 2, using âđĽ ⤠1 + đĽ and, for đ > 2, H¨older inequality to remove the powers outside the integral on the right side of (29), afterwards, one proceeds similar to handle the Lebesgue Ě3 . Combining integrals đ1 and đ2 , which leads to a constant đś đ the above discussion, one obtains óľŠ óľŠđ óľŠ óľŠđ E sup óľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ⤠4(đâ2)/2 EóľŠóľŠóľŠđĽ0 óľŠóľŠóľŠ đĄâ[đĄ0 ,đ] +
Ě1 đś đ
+
Ě2 đś đ
+
Ě3 đś đ
(31) Ěđ . â¤đś
To complete the proof one needs to show đđ â đ for đ â +â. Mentioning that the solution đĽđ(đĄ) is continuous in đĄ, thus E supđĄâ[đĄ0 ,đ] âđĽđ(đĄ ⧠đđ)âđ is bounded and converges
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5
đ-wise, for đ â +â, to E supđĄâ[đĄ0 ,đ] âđĽđ(đĄ)âđ . Moreover, Fatou Lemma and (31) imply óľŠđ óľŠ óľŠ óľŠđ E sup óľŠóľŠóľŠđĽđ (đĄ)óľŠóľŠóľŠ = E lim sup óľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ đ â +â đĄâ[đĄ0 ,đ] đĄâ[đĄ0 ,đ] óľŠ óľŠđ Ě â¤ lim inf E sup óľŠóľŠóľŠđĽđ (đĄ ⧠đđ)óľŠóľŠóľŠ ⤠đś đ. đ â +â đĄâ[đĄ0 ,đ] (32) Now, we state our first main result. Theorem 5. Let (đť1 ) and (đť2 ) be satisfied, and then (7) possesses a unique almost sure continuous solution process. Proof. It is easy to see that the uniqueness follows from the Lipschitz condition fulfilled by the coefficients (see Remark 2 in Gihman and Skorohod [31], p. 45). In the following, we only need to prove the existence of solution for (7). Note that Lemma 1 ensures the existence of a solution đĽđ(đĄ) for (11). We firstly show that đĽđ(đĄ) converges to a function đĽ0 (đĄ) for đ â +â. Again let đđ denote the stopping time đđ = inf{đĄ â [đĄ0 , đ] : âđĽđ(đĄ)â ⼠đ} for đ â N. From Chebyshev inequality and Lemma 4, it follows that { } óľŠđ óľŠ P {đđ < đ} ⤠P { sup óľŠóľŠóľŠđĽđ (đĄ)óľŠóľŠóľŠ ⼠đ} {đĄâ[đĄ0 ,đ] } óľŠóľŠđ óľŠóľŠ E (supđĄâ[đĄ0 ,đ] óľŠóľŠđĽđ (đĄ)óľŠóľŠ ) ⤠đđ Ě (đ, đ¸óľŠóľŠóľŠđĽ0 óľŠóľŠóľŠđ , đ) đś óľŠ óľŠ â¤ ół¨â 0, đđ
đĽđ(đĄ ⧠đđ) = đĽ(đĄ ⧠đđ) for đĄ ⤠đ, the almost sure convergence of đđ to đ implies óľŠóľŠ đĄ { óľŠ P { sup óľŠóľŠóľŠóľŠâŤ (đ (đĽđ (đ )) â đ (đĽ (đ ))) đđ óľŠ đĄ {đĄâ[đĄ0 ,đ] óľŠ 0 óľŠóľŠ đĄ } óľŠ + ⍠(đ (đĽđ (đ )) â đ (đĽ (đ ))) đđ (đĄ)óľŠóľŠóľŠóľŠ > 0} óľŠóľŠ đĄ0 } ⤠P {đđ < đĄ} ół¨â 0,
as đ ół¨â +â. (34)
Therefore, đĽ0 (â
) is a solution for (7) on [đĄ0 , đ]. The proof is completed. Under the assumptions (đť1 ) and (đť2 ), the solution has some important properties. Corollary 6. The solution đĽ(đĄ) for (7) has the following properties. (i) đĽ(đĄ) is a FđĄ Ă B([đĄ0 , đ])-measurable homogeneous Markov process. (ii) Let EâđĽ0 âđ < +â for a fixed even đ, and then there Ěđ = đś(đ, Ě đ¸âđĽ0 âđ , đ) such that exists a constant đś Ěđ , E sup âđĽ (đĄ)âđ ⤠âđĽ (đĄ)âđ ⤠đś đĄâ[đĄ0 ,đ]
(33)
as đ ół¨â +â. Note that this states slightly more than convergence in probability of đđ â đ. One can find, for almost every đ â Ί, an đ0 (đ) such that đđ0 (đ) = đ. Moreover, one has đđó¸ ⼠đđ đĽ đĽ and đĽđ0ó¸ (â
, đ) = đĽđ0 (â
, đ) (almost sure) on [đĄ0 , đđ] for all đó¸ ⼠đ (see Gihman and Skorohod [14], p. 4). Thus, if đđ = đ, then đđó¸ = đ for all đó¸ ⼠đ. From the above discussion, it follows that the set {đ : đđ(đ) = đ} is monotonously increasing and converges to Ί, for đ â +â. We point out once more that if đđ0 (đ) = đ, then one can express, for almost đ â Ί, the limit function by đĽ(â
, đ) = đĽđó¸ (â
, đ) for all đó¸ ⼠đ0 (đ). (Actually đĽ đĽđ0 (â
) is only a version of đĽ(â
) on [đĄ0 , đđ]; i.e., there exists an exceptional P-null set N(đ). Note that there are countable many such null sets, so that the union over all the P-null set is again a null set.) Therefore, đĽđ(đĄ) converges uniformly in đĄ to đĽ0 (đĄ), together with đĽđ(đĄ) is continuous in đĄ, đĽ0 (đĄ) is further continuous in đĄ. Secondly, we further show that the limit function đĽ0 (đĄ) is a real solution for (7). For đĄ = đĄ0 , this is true because đĽđ(đĄ0 ) = đĽ0 for all đ â N. For đĄ â (đĄ0 , đ], one considers the limit function of đĽđ(đĄ) for đ â +â. According to đđ(đĽđ(đĄ ⧠đđ)) = đ(đĽ(đĄ ⧠đđ)) and
âđĄ â [đĄ0 , đ] .
(35)
Furthermore, for every deterministic and bounded set đľ â Rđ , Ěđ is finite (đĽ0 is deterministic). the constant supđĽ0 âđľ đś Proof. (i) Because the coefficients đđ(đĽđ), đ(đĽđ) of (11) are independent of đ, đĄ respectively, and fulfill both linear growth condition and Lipschitz condition, the solution đĽđ(đĄ) is a homogeneous Markov process by Theorem 1 in Gihman and Skorohod [31] (Section 10). Therefore, the solution đĽ(đĄ) of system (7) is also a homogeneous Markov process since đĽđ(đĄ) uniformly converges to đĽ(đĄ). (ii) From the proof of Lemma 4, it is to see that there Ěđ in (ii). Note that EâđĽ(đĄ)â, đĄ â [0, đ], exists the constant đś can be bounded by a probably more accurate constant using Lemma 3. Every bounded set đľ is contained in a ball of appropriate radius đ
and center zero. Setting âđĽ0 â = đ
, the assertion follows from the linear dependence of the bounding constant (cf. (23) and (31) resp.). Corollary 7. Let 2đ2 + đ2 < 0, and then there exists a constant đś(EâđĽ0 â2 ) such that óľŠ óľŠ2 sup đ¸âđĽ (đĄ)â2 < đś (đ¸óľŠóľŠóľŠđĽ0 óľŠóľŠóľŠ ) . đĄâ(đĄ0 ,+â)
(36)
Proof. The assumptions imply that the constant đ1 in the proof of Lemma 3 is negative. Hence, there exist a bounded constant only depending on the initial condition, but not on đĄ.
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4. Applications In this section, we prove the existence and uniqueness of solution for the stochastic Lorenz system for weather forecasting as an application to illustrate the effectiveness of our results. The Lorenz system was introduced by Lorenz [34]. For the physical meaning of the system, the reader is referred to Peitgen et al. [35]. Forces not described by those equations are assumed to be random. We model these influences by white noise. This leads to the stochastic Lorenz system which has been described in detail in Arnold [36] and has been intensively studied in [6, 37] and so forth. Definition 8. Let đĽ = (đĽ, đŚ, đ§)⤠â R3 , đ˝, đ, đ > 0 be constants, then the stochastic Lorenz system is defined by đđĽ = (âđ´đĽ + đ (đĽ)) đđĄ + đ (đĽ) đđ (đĄ) ,
đĽ (đĄ0 ) = đĽ0 , (37)
the two parts of the drift are given by đ âđ 0 đ´ = (âđ 1 0 ) , 0 0 đ˝
0 đ (đĽ) = (âđĽđ§) , đĽđŚ
(38)
and the noise term đ(đĽ) : R3 â (3 Ă đ) matrix satisfies Lipschitz as well as linear growth condition. Remark 9. If one considers the noise to act in Stratonovich form, one talks about the Stratonovich Lorenz system. Note that the Stratonovich and the corresponding ItËo Lorenz system are equivalent up to an additional drift term (see Arnold [38], p. 181). Theorem 10. The stochastic Lorenz system (37) exists a unique solution. Proof. Let đĽó¸ = đĽ, đŚó¸ = đŚ, đ§ó¸ = đ§âđâđ, and đ = (đĽó¸ , đŚó¸ , đ§ó¸ )⤠â R3 , and one has đđ = (âđˇđ â đľ + đ (đ)) đđĄ + đ (đ) đđ (đĄ) .
(39)
The three parts of the drift are given by 0 đ (đ) = (âđĽó¸ đ§ó¸ ) , đĽ ó¸ đŚó¸
đ âđ 0 đˇ = (đ 1 0 ) , 0 0 đ˝
0 đľ = ( 0 ). đ˝ (đ + đ)
(40)
Obviously, the noise term đ(đ) : R3 â (3 Ă đ) matrix remains satisfying Lipschitz as well as linear growth condition. According to (40), one has â¨đ(đ), đ⊠= 0. Moreover, 2
2
2
â¨âđˇđ â đľ, đ⊠= âđđĽó¸ â đŚó¸ â đ˝đ§ó¸ â đ˝ (đ + đ) đ§ó¸ đ˝2 (đ + đ)2 1 ⤠(âđ + ) âđâ2 + , 2 2
(41)
where đ = min{đ, 1, đ˝}. Hence according to Theorem 10, system (39) exists a unique solution, so is system (37).
5. Conclusion This paper considers the problems of existence and uniqueness of solution for a class of stochastic differential equations whose nonlinear part does not satisfy linear growth condition. Some new criteria ensuring the existence and uniqueness of solution were presented. These criteria extend, improve, complement a number of results about the existence and uniqueness of solution, and handle some cases not covered by known criteria. Furthermore, these criteria are important in applications such as stochastic Lorenz system for weather forecasting, some other systems, and so on.
Acknowledgments This work is supported by the National Natural Science Foundation of China (no. 11271139), (no. 11201089) and (no. 11301090) and Guangxi Natural Science Foundation (no. 2013GXNSFAA019014) and (no. 2013GXNSFBA019016).
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