Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations

Abstract

A class of stochastic differential equations given by 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡))𝑑𝑡+𝑔(𝑥(𝑡))𝑑𝑊(𝑡),  𝑥(𝑡₀)=𝑥₀,  𝑡₀≤𝑡≤𝑇<+∞, 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 [1–9] 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 [10–14] and the references therein. Especially, Mao had investigated the stochastic differential equations: 𝑑𝑋(𝑡)=𝑓(𝑋(𝑡),𝑡)𝑑𝑡+𝑔(𝑋(𝑡),𝑡)𝑑𝐵_𝑡,𝑋𝑡₀=𝑋₀, (1) on the closed interval [𝑡₀,𝑇], 𝑡₀≤𝑇, in his book [14], and obtained that if Lipschitz condition 𝑓(𝑋,𝑡)−𝑓𝑋,𝑡²∨𝑔(𝑋,𝑡)−𝑔𝑋,𝑡²≤𝐾𝑋−𝑋²,𝑋,𝑋∈ℝ^𝑑,𝑡∈𝑡₀,𝑇, (2) and linear growth condition: 𝑓(𝑋,𝑡)²∨𝑔(𝑋,𝑡)²≤𝐾1+‖𝑋‖²,(𝑋,𝑡)∈ℝ^𝑑×𝑡₀,𝑇, (3) hold, then (1) had a unique solution 𝑋(𝑡) satisfying 𝑋(𝑡)∈ℳ²([𝑡₀,𝑇],ℝ^𝑑). Furthermore, Mao [14] also discussed stochastic functional differential equations with finite delay: 𝑑𝑋(𝑡)=𝑓𝑋_𝑡,𝑡𝑑𝑡+𝑔𝑋_𝑡,𝑡𝑑𝐵_𝑡,𝑋_𝑡0=𝜉,𝑡₀≤𝑡≤𝑇<+∞, (4) where 𝑋_𝑡={𝑋(𝑡+𝜃)∶−𝜏≤𝜃≤0} could be considered as a 𝐶([𝑡₀,𝑇],ℝ^𝑑)-value stochastic process. The initial value 𝑋_𝑡0=𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0} is an 𝐹_𝑡0-measurable 𝐶([𝑡₀,𝑇],ℝ^𝑑)-value random variable such that 𝔼‖𝜉‖²<∞. For (4), if uniform Lipschitz condition 𝑓𝜑,𝑡−𝑓𝜙,𝑡²∨𝑔𝜑,𝑡−𝑔𝜙,𝑡²≤𝐾𝜑−𝜙²,𝐾>0, (5) for any 𝜑,𝜙∈𝐶([𝑡₀,𝑇];ℝ^𝑑), 𝑡∈[𝑡₀,𝑇], and linear growth condition 𝑓𝜑,𝑡²∨𝑔𝜑,𝑡²≤𝐾1+𝜑²,𝜑,𝑡∈ℝ^𝑑×𝑡₀,𝑇,𝐾>0, (6) are satisfied, then (4) had a unique solution 𝑋(𝑡); moreover, 𝑋(𝑡)∈ℳ²([𝑡₀,𝑇],ℝ^𝑑).

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.

2. Preliminaries

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: ℝ^𝑛 denotes the usual 𝑛-dimensional Euclidean space, ‖⋅‖ denotes norm in ℝ^𝑛. If 𝐴 is a vector or a matrix, its transpose is denoted by 𝐴^𝑇; if 𝐴 is a matrix, its trace norm is represented by ‖⋅‖=trace(𝐴^𝑇𝐴). Let (Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_{𝑡∈[𝑡0,+∞)} satisfying the usual conditions (i.e., it is increasing and right continuous). ℱ_𝑡0 is independent of the 𝜎-field generated by {𝑊(𝑡)−𝑊(𝑡₀)∶𝑡₀≤𝑡≤𝑇} and contains all ℙ-null sets. 𝑊(𝑡) is a given 𝑚-dimensional standard Brownian motion.

Consider 𝑛-dimensional stochastic differential equations: (𝑡)(𝑥(𝑡(𝑥(𝑡(𝑡),𝑥𝑡𝑑𝑥=𝑓))𝑑𝑡+𝑔))𝑑𝑊₀=𝑥₀,𝑡₀≤𝑡≤𝑇<+∞, (7) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^⊤, 𝑥₀=(𝑥₁(𝑡₀),𝑥₂(𝑡₀),…,𝑥_𝑛(𝑡₀))^⊤ is independent of ℱ_𝑡, and satisfies 𝔼‖𝑥₀‖²<+∞,𝑊(𝑡) is an 𝑚-dimensional Wiener process, 𝑓(𝑥(𝑡))=(𝑓₁(𝑥(𝑡)),𝑓₂(𝑥(𝑡)),…𝑓_𝑛(𝑥(𝑡)))^⊤, 𝑔(𝑥(𝑡)) is a 𝑛×𝑚 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.

(𝐻₁)𝑓(𝑥(𝑡)) satisfies the Lipschitz condition; moreover, 𝑓(𝑥),𝑥≤𝑘₁+𝑘₂‖𝑥‖²; (8)

(𝐻₂)𝑔(𝑥(𝑡)) satisfies the Lipschitz condition and the linear growth condition: 𝑔(𝑥)²≤𝑘²1+‖𝑥‖², (9) where ⟨⋅,⋅⟩ denotes the inner product of ℝ^𝑛 and 𝑘₁, 𝑘₂, and 𝑘 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 (𝐻₁) and (𝐻₂) be satisfied. Setting 𝜒_𝑁∈𝐶¹(ℝ^𝑛,ℝ) with 𝜒_𝑁(𝑥)=‖𝑥‖‖𝑥‖1,𝑓𝑜𝑟≤𝑁,0,𝑓𝑜𝑟≥𝑁+1, (10) and 𝑓_𝑁(𝑥)=𝜒_𝑁(𝑥)𝑓(𝑥), and then the modified stochastic differential equations: 𝑑𝑥_𝑁=𝑓_𝑁𝑥_𝑁𝑑𝑡+𝑔𝑥_𝑁𝑑𝑊_𝑡,𝑥_𝑁𝑡₀=𝑥₀,𝑡₀≤𝑡≤𝑇<+∞, (11) possesses a continuous almost sure unique ℱ_𝑡 measurable solution process.

Proof. As the truncation function 𝜒_𝑁∈𝐶¹(ℝ^𝑛,ℝ), 𝑓_𝑁(𝑥) remains differentiable and its derivative is both continuous and has a compact support. Hence 𝑓_𝑁(𝑥) is bounded and satisfies linear growth condition as well as Lipschitz condition. According to (𝐻₂), 𝑔(𝑥) satisfies linear growth condition and Lipschitz condition. Therefore, the assertion follows by the usual existence and uniqueness theorem (see Gihman and Skorohod [31], p. 37–47).

Lemma 2 (see [32]). Let 𝐶, 𝐷 be both positive and semidefinite Hermite matrix, and 𝑘 is a positive integer; then trace(𝐶𝐷)^𝑘≤(trace(𝐶))^𝑘(trace(𝐷))^𝑘. (12)

We compute the 𝑝th norm of the solution for (11) as follows. By Itô formula w.r.t. the function ℎ(𝑥)=‖𝑥‖^𝑝=(∑^𝑛_𝑖=1𝑥²_𝑖)^𝑝/2 for 𝑝 being even, one obtains 𝑑𝑥_𝑁^𝑝=𝑝𝑥_𝑁^𝑝−2𝑓𝑥_𝑁,𝑥_𝑁𝑝𝑑𝑡+𝑝2−1𝑥_𝑁^𝑝−4trace𝑥_𝑁𝑥^𝑇_𝑁𝑔𝑥_𝑁×𝑔^𝑇𝑥_𝑁+𝑝𝑑𝑡2𝑥_𝑁^𝑝−2trace𝑔𝑥_𝑁𝑔^𝑇𝑥_𝑁𝑥𝑑𝑡+𝑝_𝑁^𝑝−2𝑥^𝑇_𝑁𝑔𝑥_𝑁𝑑𝑊(𝑡). (13) According to Lemma 2, one has trace𝑥^𝑇_𝑁𝑥_𝑁𝑔^𝑇𝑥_𝑁𝑔𝑥_𝑁≤𝑥_𝑁²𝑔𝑥_𝑁². (14) Together with (𝐻₁), the differential of the 𝑝th norm is given by 𝑑𝑥_𝑁^𝑝=𝑝𝑘₁𝑥_𝑁^𝑝−2+𝑝𝑘₂𝑥_𝑁^𝑝+𝑝𝑑𝑡𝑝−12𝑥_𝑁^𝑝−2𝑔𝑥_𝑁²𝑥𝑑𝑡+𝑝_𝑁^𝑝−2𝑥^𝑇_𝑁𝑔𝑥_𝑁𝑑𝑊(𝑡)+𝜉(𝑡)𝑑𝑡, (15) where 𝜉(𝑡)≤0 is an adapted process that compensates all the estimation made and could be computed explicitly.

Lemma 3. Let 𝑝∈ℕ be even, and then there exists a constant 𝐶_𝑝=𝐶(𝑇,𝐸‖𝑥₀‖^𝑝,𝑝) independent of 𝑁 such that 𝐸‖𝑥_𝑁‖^𝑝<𝐶_𝑝 for 𝑡∈[𝑡₀,𝑇].

Proof. For 𝑀∈ℕ, define stopping time 𝜏_𝑀=inf{𝑡∈[𝑡₀,𝑇]∶‖𝑥_𝑁(𝑡)‖≥𝑀}. Note that ^{𝑡∧𝜏𝑀}_𝑡0𝑓(𝑠)𝑑𝑠≤^𝑡_𝑡0𝑓𝑠∧𝜏_𝑀𝑑𝑠for𝑓(𝑡)≥0. (16) Since ‖𝑥_𝑁(𝑡∧𝜏_𝑀)‖^𝑞≥𝑀^𝑞 for all 𝑞>0, using (𝐻₂) and the above inequality, one concludes that integrand of the stochastic integral is bounded; hence, 𝔼^{𝑡∧𝜏𝑀}_𝑡0𝑝𝑥_𝑁(𝑠)^𝑝−2𝑥_𝑁(𝑠)^𝑇𝑔𝑥_𝑁(𝑠)𝑑𝑊(𝑠)=0. (17) Form (𝐻₂), we get the following inequality: 𝔼𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤𝔼𝑥𝑡₀^𝑝+^𝑡_𝑡0𝑝𝑘₁+𝑘²𝑝𝑝−12×𝔼𝑥_𝑁𝑠∧𝜏_𝑀^𝑝−2+𝑝𝑘₂+𝑘²𝑝𝑝−12×𝔼𝑥_𝑁𝑠∧𝜏_𝑀^𝑝𝑑𝑠. (18) Let 𝑑₁=𝑝𝑘₂+𝑘²𝑝(𝑝−1)/2, 𝑑₂=𝑝𝑘₁+𝑘²𝑝(𝑝−1)/2, then (18) becomes 𝔼𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤𝔼𝑥𝑡₀^𝑝+^𝑡_𝑡0𝑑₁𝔼𝑥_𝑁𝑡∧𝜏_𝑀^𝑝+𝑑₂𝔼𝑥_𝑁𝑡∧𝜏_𝑀^𝑝−2𝑑𝑠, (19) which implies 𝔼𝑥_𝑁𝑡∧𝜏_𝑀²≤𝔼𝑥𝑡₀²+^𝑡_𝑡0𝑑₁𝔼𝑥_𝑁𝑠∧𝜏_𝑀²+𝑑₂𝑑𝑠. (20) Gronwall inequality implies that there exists a constant 𝐶₂ such that sup_{𝑡𝑡∈0,𝑇}𝔼𝑥_𝑁𝑠∧𝜏_𝑀²≤𝐶₂. (21) By inductive method, there exist a constant 𝐶_𝑝 such that 𝔼𝑥_𝑁𝑠∧𝜏_𝑀^𝑝≤𝔼𝑥𝑡₀^𝑝+^𝑡_𝑡0𝑑₁𝔼𝑥_𝑁𝑠∧𝜏_𝑀^𝑝+𝑑₂𝐶_𝑝−2𝑑𝑠≤𝐶_𝑝. (22) It remains to show that the stopping time satisfies 𝜏_𝑀→𝑇 for 𝑀→+∞. By the continuity of the solution 𝑥_𝑁(𝑡) in 𝑡, the norm ‖𝑥_𝑁(𝑡∧𝜏_𝑀)‖^𝑝 is bounded; therefore, it converges 𝜔-wise to ‖𝑥_𝑁(𝑡)‖^𝑝 as 𝑀→+∞. Since the norm is nonnegative and bounded for all 𝑀∈ℕ, for 𝑡≤𝑇, Fatou Lemma implies that 𝔼𝑥_𝑁(𝑡)^𝑝≤𝔼lim_𝑀→+∞𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤liminf_𝑀→+∞𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤𝐶_𝑝. (23)

Lemma 4. Let 𝑝∈ℕ be even, and then there exists a constant 𝐶_𝑝=𝐶(𝑇,𝐸‖𝑥₀‖^𝑝,𝑝) independent of 𝑁 such that 𝔼sup_{𝑡∈[𝑡0,𝑇]}‖𝑥_𝑁(𝑡)‖^𝑝<𝐶_𝑝 for 𝑡∈[𝑡₀,𝑇].

Proof. From (15), it follows that 𝑥_𝑁𝑡∧𝜏_𝑀²=𝑥𝑡₀²+^{𝑡∧𝜏𝑀}_𝑡02𝑘₂𝑥_𝑁(𝑠)²+𝑑𝑠_{𝑡∧𝜏𝑀𝑡0}2𝑘₁+𝑔𝑥_𝑁(𝑠)²+𝑑𝑠_{𝑡∧𝜏M𝑡0}2𝑥^𝑇_𝑁(𝑠)𝑔𝑥_𝑁(𝑠)𝑑𝑊(𝑠)+^{𝑡∧𝜏𝑀}_𝑡0𝜉(𝑠)𝑑𝑠, (24) which implies 𝑥_𝑁𝑡∧𝜏_𝑀²≤𝑥𝑡₀²+^{𝑡∧𝜏𝑀}_𝑡02𝑘₂𝑥_𝑁(𝑠)²+𝑑𝑠_{𝑡∧𝜏𝑀𝑡0}2𝑘₁+𝑔𝑥_𝑁(𝑠)²+𝑑𝑠_{𝑡∧𝜏𝑀𝑡0}2𝑥^𝑇_𝑁(𝑠)𝑔𝑥_𝑁(𝑠)𝑑W(𝑠). (25) Taking to the power 𝑝/2 to obtain an expression for ‖𝑥_𝑁(𝑡∧𝜏_𝑀)‖^𝑝, one has 𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤𝑥𝑡₀²+^{𝑡∧𝜏𝑀}₀2𝑘₂𝑥_𝑁(𝑠)²+𝑑𝑠_{𝑡∧𝜏𝑀𝑡0}2𝑘₁+𝑔𝑥_𝑁(𝑠)²+𝑑𝑠_{𝑡∧𝜏𝑀𝑡0}2𝑥^𝑇_𝑁(𝑠)𝑔𝑥_𝑁(𝑠)𝑑𝑊(𝑠)^𝑝/2. (26) According to |∑^𝑘_𝑖=1𝑎_𝑖|^𝑚≤𝑘^𝑚−1∑^𝑘_𝑖=1|𝑎_𝑖|^𝑚 for 𝑚≥1, one obtains 𝔼sup_{𝑡𝑡∈0,𝑇}𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤4^(𝑝−2)/2𝔼𝑥𝑡₀^𝑝+4^(𝑝−2)/2𝔼^𝑇_𝑡02𝑘₂𝑥_𝑁(𝑠)²𝑑𝑠^𝑝/2+4^(𝑝−2)/2𝔼^𝑇_𝑡02𝑘₁+𝑔𝑥_𝑁(𝑠)²𝑑𝑠^𝑝/2+4^(𝑝−2)/2𝔼sup_{𝑡𝑡∈0,𝑇}^{𝑡∧𝜏𝑀}_𝑡02𝑥^𝑇_𝑁(𝑠)𝑔𝑥_𝑁(𝑠)𝑑𝑊(𝑠)^𝑝/2=4^(𝑝−2)/2𝐸𝑥𝑡₀^𝑝+𝑈₁+𝑈₂+𝑈₃. (27) Since the solution 𝑥_𝑁(𝑡) for (11) is ℱ_𝑡 measurable for 𝑡≥𝑡₀ and continuous in 𝑡, the norm ‖𝑥_𝑁(𝑡)‖ is ℱ_𝑡⊗ℬ([𝑡₀,𝑡])-measurable (see Wentzell [33], p. 89, p. 18). Therefore, applying (𝐻₂), together with Hölder inequality to remove the powers outside the deterministic integrals 𝑈₁ and 𝑈₂, changing expectation and integration by Fubini theorem and using Lemma 3, one obtains 𝑈₁=4^(𝑝−2)/2𝔼^𝑇_𝑡02𝑘₂𝑥_𝑁(𝑠)²𝑑𝑠^𝑝/2≤𝐶¹_𝑝,𝑈₂=4^(𝑝−2)/2𝔼^𝑇_𝑡02𝑘₁+𝑔𝑥_𝑁(𝑠)²𝑑𝑠^𝑝/2≤4^(𝑝−2)/2𝑇^(𝑝−2)/2×^𝑇_𝑡02𝑘₁^𝑝/2+𝑘^𝑝+𝑘^𝑝𝐶₂𝑑𝑠, (28) which is bounded by a constant 𝐶²_𝑝. To further estimate the stochastic integral 𝑈₃, using the Burkholder-Davis-Gundy (see Mao [14], p. 7), one obtains 𝑈₃=𝔼sup_{𝑡𝑡∈0,𝑇}^𝑡_𝑡0𝑥^𝑇_𝑁(𝑠)𝑔𝑥_𝑁(𝑠)𝑑𝑊(𝑠)^𝑝/2≤𝑘_𝑝𝔼^𝑇_𝑡0𝑥_𝑁(𝑠)𝑔𝑥_𝑁(𝑠)^2𝑝/4𝑑𝑠, (29) with the constant 𝑘_𝑝=64𝑝^𝑝/4𝑝,for0<𝑝<4,_𝑝/2+12^𝑝/2+2𝑝/2−1^{𝑝/2−1𝑝/4},for𝑝≥4. (30) For 𝑝=2, using √𝑥≤1+𝑥 and, for 𝑝>2, Hölder inequality to remove the powers outside the integral on the right side of (29), afterwards, one proceeds similar to handle the Lebesgue integrals 𝑈₁ and 𝑈₂, which leads to a constant 𝐶³_𝑝. Combining the above discussion, one obtains 𝔼sup_{𝑡𝑡∈0,𝑇}𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤4^(𝑝−2)/2𝔼𝑥₀^𝑝+𝐶¹_𝑝+𝐶²_𝑝+𝐶³_𝑝≤𝐶_𝑝. (31) To complete the proof one needs to show 𝜏_𝑀→𝑇 for 𝑀→+∞. Mentioning that the solution 𝑥_𝑁(𝑡) is continuous in 𝑡, thus 𝔼sup_{𝑡∈[𝑡0,𝑇]}‖𝑥_𝑁(𝑡∧𝜏_𝑀)‖^𝑝 is bounded and converges 𝜔-wise, for 𝑀→+∞, to 𝔼sup_{𝑡∈[𝑡0,𝑇]}‖𝑥_𝑁(𝑡)‖^𝑝. Moreover, Fatou Lemma and (31) imply 𝔼sup_{𝑡𝑡∈0,𝑇}𝑥_𝑁(𝑡)^𝑝=𝔼lim_𝑀→+∞sup_{𝑡𝑡∈0,𝑇}𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤liminf_𝑀→+∞𝔼sup_{𝑡𝑡∈0,𝑇}𝑥_𝑁𝑡∧𝜏_𝑀^𝑝≤𝐶_𝑝. (32)
Now, we state our first main result.

Theorem 5. Let (𝐻₁) and (𝐻₂) 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 𝑥_𝑁(𝑡) for (11). We firstly show that 𝑥_𝑁(𝑡) converges to a function 𝑥⁰(𝑡) for 𝑁→+∞.
Again let 𝜏_𝑁 denote the stopping time 𝜏_𝑁=inf{𝑡∈[𝑡₀,𝑇]∶‖𝑥_𝑁(𝑡)‖≥𝑁} for 𝑁∈ℕ. From Chebyshev inequality and Lemma 4, it follows that ℙ𝜏_𝑁<𝑇≤ℙsup_{𝑡𝑡∈0,𝑇}𝑥_𝑁(𝑡)^𝑝≤𝔼≥𝑁sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑁(𝑡)^𝑝𝑁^𝑃≤𝐶𝑇,𝐸𝑥₀^𝑝,𝑝𝑁^𝑃⟶0,as𝑁⟶+∞. (33) Note that this states slightly more than convergence in probability of 𝜏_𝑁→𝑇. One can find, for almost every 𝜔∈Ω, an 𝑁₀(𝜔) such that 𝜏_𝑁0(𝜔)=𝑇. Moreover, one has 𝜏_𝑁′≥𝜏_𝑁 and 𝑥^𝑥0_𝑁′(⋅,𝜔)=𝑥^𝑥0_𝑁(⋅,𝜔) (almost sure) on [𝑡₀,𝜏_𝑁] for all 𝑁^′≥𝑁 (see Gihman and Skorohod [14], p. 4). Thus, if 𝜏_𝑁=𝑇, then 𝜏_𝑁′=𝑇 for all 𝑁^′≥𝑁. From the above discussion, it follows that the set {𝜔∶𝜏_𝑁(𝜔)=𝑇} is monotonously increasing and converges to Ω, for 𝑁→+∞. We point out once more that if 𝜏_𝑁0(𝜔)=𝑇, then one can express, for almost 𝜔∈Ω, the limit function by 𝑥(⋅,𝜔)=𝑥_𝑁′(⋅,𝜔) for all 𝑁^′≥𝑁₀(𝜔). (Actually 𝑥^𝑥0_𝑁(⋅) is only a version of 𝑥(⋅) on [𝑡₀,𝜏_𝑁]; i.e., there exists an exceptional ℙ-null set 𝒩(𝑁). 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, 𝑥_𝑁(𝑡) converges uniformly in 𝑡 to 𝑥⁰(𝑡), together with 𝑥_𝑁(𝑡) is continuous in 𝑡, 𝑥⁰(𝑡) is further continuous in 𝑡.
Secondly, we further show that the limit function 𝑥⁰(𝑡) is a real solution for (7).
For 𝑡=𝑡₀, this is true because 𝑥_𝑁(𝑡₀)=𝑥₀ for all 𝑁∈ℕ. For 𝑡∈(𝑡₀,𝑇], one considers the limit function of 𝑥_𝑁(𝑡) for 𝑁→+∞. According to 𝑓_𝑁(𝑥_𝑁(𝑡∧𝜏_𝑁))=𝑓(𝑥(𝑡∧𝜏_𝑁)) and 𝑥_𝑁(𝑡∧𝜏_𝑁)=𝑥(𝑡∧𝜏_𝑁) for 𝑡≤𝑇, the almost sure convergence of 𝜏_𝑁 to 𝑇 implies ℙsup_{𝑡𝑡∈0,𝑇}^𝑡_𝑡0𝑓𝑥_𝑁(𝑠)(𝑥(𝑠+−𝑓))𝑑𝑠_𝑡𝑡0𝑔𝑥_𝑁(𝑠)(𝑥(𝑠(𝑡𝜏−𝑔))𝑑𝑊)>0≤ℙ_𝑁<𝑡⟶0,as𝑁⟶+∞. (34) Therefore, 𝑥⁰(⋅) is a solution for (7) on [𝑡₀,𝑇]. The proof is completed.

Under the assumptions (𝐻₁) and (𝐻₂), the solution has some important properties.

Corollary 6. The solution 𝑥(𝑡) for (7) has the following properties.(i)𝑥(𝑡) is a ℱ_𝑡×ℬ([𝑡₀,𝑇])-measurable homogeneous Markov process.(ii)Let 𝔼‖𝑥₀‖^𝑝<+∞ for a fixed even 𝑝, and then there exists a constant 𝐶_𝑝=𝐶(𝑇,𝐸‖𝑥₀‖^𝑝,𝑝) such that 𝔼sup_{𝑡𝑡∈0,𝑇}‖𝑥(𝑡)‖^𝑝≤‖𝑥(𝑡)‖^𝑝≤𝐶_𝑝,∀𝑡∈𝑡₀,𝑇. (35)Furthermore, for every deterministic and bounded set 𝐵⊂ℝ^𝑛, the constant sup_𝑥0∈𝐵𝐶_𝑝 is finite (𝑥₀ is deterministic).

Proof. (i) Because the coefficients 𝑓_𝑁(𝑥_𝑁), 𝑔(𝑥_𝑁) of (11) are independent of 𝜔, 𝑡 respectively, and fulfill both linear growth condition and Lipschitz condition, the solution 𝑥_𝑁(𝑡) is a homogeneous Markov process by Theorem 1 in Gihman and Skorohod [31] (Section 10). Therefore, the solution 𝑥(𝑡) of system (7) is also a homogeneous Markov process since 𝑥_𝑁(𝑡) uniformly converges to 𝑥(𝑡).

(ii) From the proof of Lemma 4, it is to see that there exists the constant 𝐶_𝑝 in (ii). Note that 𝔼‖𝑥(𝑡)‖, 𝑡∈[0,𝑇], can be bounded by a probably more accurate constant using Lemma 3. Every bounded set 𝐵 is contained in a ball of appropriate radius 𝑅 and center zero. Setting ‖𝑥₀‖=𝑅, the assertion follows from the linear dependence of the bounding constant (cf. (23) and (31) resp.).

Corollary 7. Let 2𝑘₂+𝑘²<0, and then there exists a constant 𝐶(𝔼‖𝑥₀‖²) such that sup_{𝑡𝑡∈0,+∞}𝐸‖𝑥(𝑡)‖²<𝐶𝐸𝑥₀². (36)

Proof. The assumptions imply that the constant 𝑑₁ in the proof of Lemma 3 is negative. Hence, there exist a bounded constant only depending on the initial condition, but not on 𝑡.

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 𝑥=(𝑥,𝑦,𝑧)^⊤∈ℝ³, 𝛽,𝑟,𝜎>0 be constants, then the stochastic Lorenz system is defined by 𝑑𝑥=−𝐴𝑥+𝑓(𝑥)𝑑𝑡+𝑔(𝑥)𝑑𝑊(𝑡),𝑥𝑡₀=𝑥₀, (37) the two parts of the drift are given by (𝑥)=0𝐴=𝜎−𝜎0−𝑟1000𝛽,𝑓−𝑥𝑧𝑥𝑦, (38) and the noise term 𝑔(𝑥)∶ℝ³→(3×𝑚) 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 Itô 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 𝑥^′=𝑥, 𝑦^′=𝑦, 𝑧^′=𝑧−𝑟−𝜎, and 𝜔=(𝑥^′,𝑦^′,𝑧^′)^⊤∈ℝ³, and one has 𝑑𝜔=−𝐷𝜔−𝐵+𝑓(𝜔)𝑑𝑡+𝑔(𝜔)𝑑𝑊(𝑡). (39) The three parts of the drift are given by 𝐷=,𝜎−𝜎0𝜎1000𝛽𝑓(𝜔)=0−𝑥^′𝑧^′𝑥^′𝑦^′,00𝛽()𝐵=.𝑟+𝜎 (40) Obviously, the noise term 𝑔(𝜔)∶ℝ³→(3×𝑚) matrix remains satisfying Lipschitz as well as linear growth condition. According to (40), one has ⟨𝑓(𝜔),𝜔⟩=0. Moreover, ⟨−𝐷𝜔−𝐵,𝜔⟩=−𝜎𝑥^′2−𝑦^′2−𝛽𝑧^′2−𝛽(𝑟+𝜎)𝑧^′≤1−𝑎+2‖𝜔‖²+𝛽²(𝑟+𝜎)²2, (41) where 𝑎=min{𝜎,1,𝛽}. 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).