非Lipschitz条件下倒向随机微分方程解的稳定性

非Lipschitz条件下双重倒向随机微分方程的适应解和比较定
理
段鹏举;张祖峰
【期刊名称】《宿州学院学报》
【年(卷),期】2011(26)5
【摘要】在非Lipschitz条件下,通过构造Picard逼近序列,研究了一类由Kunita-Ito积分驱动的双重倒向随机微分方程解的存在唯一性,从而弱化了方程解的存在唯一性条件,并且在此非Lipschitz条件下,进一步讨论了方程解的性质,也就是方程解的比较定理.
【总页数】6页(P5-9,52)
【作者】段鹏举;张祖峰
【作者单位】宿州学院数学系,安徽宿州,234000;宿州学院智能信息处理实验室,安徽宿州,234000;宿州学院数学系,安徽宿州,234000
【正文语种】中文
【中图分类】O211.6
【相关文献】
1.非Lipschitz条件下带扰动倒向随机微分方程的比较定理 [J], 孙丹丹
2.非Lipschitz条件下反射倒向随机微分方程解的性质 [J], 梁青
3.非Lipschitz条件下带跳的倒向随机微分方程的比较定理 [J], 韩宝燕
4.非Lipschitz条件下倒向随机微分方程的比较定理 [J], 韩宝燕;朱波
5.非Lipschitz条件下倒向随机微分方程的比较定理(英文) [J], 孙信秀
因版权原因，仅展示原文概要，查看原文内容请购买。

非线性微分方程解的稳定性非线性微分方程解的稳定性是数学物理等多个学科面对微分方程解时所要考虑的重要问题。

一、非线性微分方程解的稳定性1. 含有稳定性的概念非线性微分方程求解的稳定性是指改变求解方法或迭代步长时，得到的求解结果的差异是限定的范围，从而确定所使用的解法或迭代过程的可靠性。

2. 非线性微分方程求解的稳定性判断求解非线性微分方程的稳定性主要判断其所使用的解法的收敛性以及使用的迭代步长的可靠性。

二、影响非线性微分方程解稳定性的因素1. 微分方程本身特征由于求解非线性微分方程的过程是多参数的复杂迭代运算，它本身的复杂性也影响了求解的稳定性。

如方程的阶数较高、参数较多等，它们会加大求解过程的难度，影响对结果的准确性及稳定性。

2. 求解方法的限制由于当下的求解方法还不能充分支撑求解非线性微分方程解过程，因而会造成求解结果的不稳定性。

3. 天气因素除了方程本身及求解方法等原因之外，天气因素也会直接影响非线性微分方程求解的稳定性，对天气变化的相关参数实时的监测和分析，及时调整迭代过程的参数设置，也是影响求解稳定性的一个重要因素。

三、维持非线性微分方程解稳定性1. 加强数值分析求解非线性微分方程时可以使用更加先进、准确的数值分析技术，分析问题的不确定性等，进行参数预估，从而可以稳定微分方程求解的结果。

2. 针对性修改求解方法多种求解方法可以在一定程度上修正或调节求解结果的不稳定性，以及减轻重要的误差，从而避免非线性微分方程求解的稳定性出现明显的变化。

3. 建立状态变化分析模型根据各参数的变化和影响，建立状态变化分析模型，可以更好地把握系统的运行情况变化，从而保证非线性微分方程解的稳定性。

四、总结微分方程求解的稳定性是指求解结果随参数变化或求解方法变化的差异，其稳定性的确定及提高是面对此类问题必须认真考虑的，应通过加强数值分析，针对性修改求解方法，建立状态变化分析模型等多种方法，以确保非线性微分方程求解的稳定性及准确性。

目录摘要 (3)ABSTRACT (4)前言 (5)微分方程稳定性分析原理 (6)捕鱼业的持续收获模型 (10)种群的相互竞争模型 (14)参考文献 (18)摘要微分方程稳定性理论是微分方程的一个重要的理论。

微分方程理论就是通过一些定量的计算来研究系统的稳定性，也就是系统在受到干扰项偏离平衡状态后能否恢复到平衡状态或者是平衡状态附近的位置。

用微分方程描述的物质运动的特点依赖于初值，而初值的计算或者测定不可避免的又会出现误差和干扰。

如果描述这个系统运动的微分方程的特解是不稳定的，则初值的微小误差和干扰都会导致严重的后果。

因此，不稳定的特解不适合作为我们研究问题的依据，只有稳定的特解才是我们需要的。

本文就一阶微分方程和二阶微分方程的平衡点及稳定性进行了分析，并且建立了捕鱼业持续收获模型和两种群相互竞争模型。

【关键词】微分方程；平衡点；稳定性；数学建模ABSTRACTDifferential equation stability theory is an important theory of differential equations. Differential equation theory is to study the stability of the system by some quantitative calculation, also is the system in the disturbance of deviating from the equilibrium state after the item will return to equilibrium or is near the equilibrium position. Using differential equation to describe the characteristics of the material movement depends on the initial value, and the calculation of initial value or determination of the inevitable will appear the error and interference. If the special solution of the differential equation describing the system movement is unstable, the initial value of small errors and interference will lead to serious consequences.Therefore, special solution is not suitable for the unstable as the basis of our research question, only stable solution is we need. In this paper, the first order differential equation of second order differential equation and the balance and the stability are analyzed, and the fishing sustained yield model is established and two species and two species competing models.【key words】Differential equations; Balance; Stability; Mathematical modeling前言在现实世界里，无论是在自然科学或者是社会科学的各领域中，存在着许许多多的变化规律可以用某些特定的数学模型来进行描述。

常微分方程的存在唯一性与稳定性存在唯一性与稳定性是常微分方程研究中的重要问题。

在本文中，我们将探讨常微分方程存在唯一解的条件以及解的稳定性。

一、常微分方程的存在唯一性常微分方程描述了一个未知函数及其导数之间的关系。

对于形如dy/dx = f(x, y)的一阶常微分方程，其中y是未知函数，x是自变量，f是已知函数，我们来讨论方程的存在唯一性。

1. 狄利克雷条件（Dini定理）狄利克雷条件是常微分方程存在唯一解的充分条件之一。

具体而言，如果在所考虑的区域上，函数f(x, y)连续且关于y满足Lipschitz条件，则常微分方程dy/dx = f(x, y)在该区域上存在唯一解。

2. 古典解与强解对于一阶常微分方程，如果解y的导数也是函数x的连续函数，则称该解为古典解。

如果解y满足方程dy/dx = f(x, y)，且在给定的初始条件下，解在某一区间上存在且唯一，则称该解为强解。

3. 积分常数的任意性在某些情况下，常微分方程的解不是唯一的，而是存在积分常数。

这意味着在通解中会出现某个常数，而不同的常数取值将对应不同的特解。

二、常微分方程的稳定性稳定性是指在微小扰动下，解是否保持不变或趋于某个特定值。

常微分方程的稳定性可以分为以下几种情况：1. 渐近稳定性如果对于一个常微分方程的解，当自变量趋于无穷大时，解趋于某个有界值，则称该解为渐近稳定解。

2. 指数稳定性如果对于一个常微分方程的解，存在一个常数K和正数C，使得解的绝对值小于Ce^Kx，则称该解为指数稳定解。

3. Lyapunov稳定性Lyapunov稳定性是一种更加一般化的稳定性概念。

它涉及到一个称为Lyapunov函数的函数，通过对该函数的变化率进行研究来判断解的稳定性。

总之，常微分方程的存在唯一性与稳定性是常微分方程理论中的重要研究内容。

通过适当的条件和方法，我们可以确定常微分方程的解的存在性，并对解的稳定性进行分析。

这对于解决实际问题和理解动态系统的行为具有重要意义。

微分方程的稳定性与解存在性分析在数学领域中，微分方程是研究物理、工程、经济和生物等领域中数学建模的一种重要工具。

微分方程的稳定性和解的存在性是微分方程理论中的核心概念。

本文将对微分方程的稳定性和解的存在性进行分析。

一、微分方程的稳定性分析微分方程的稳定性描述了解的行为在不同条件下的稳定情况。

稳定性的分析通常包括平衡点的稳定性和解的稳定性两个方面。

1. 平衡点的稳定性平衡点是微分方程中解保持不变的点。

考虑一个一阶常微分方程dy/dt=f(y)，当f(y)=0时，y的值处于平衡点。

为了判断平衡点的稳定性，有以下三种情况：a) 当f'(y)<0时，该平衡点是稳定的。

意味着当y离开平衡点时，解会回到平衡点附近。

b) 当f'(y)>0时，该平衡点是不稳定的。

当y离开平衡点时，解将远离平衡点。

c) 当f'(y)=0时，无法确定平衡点的稳定性，需要进行进一步的分析。

2. 解的稳定性除了平衡点的稳定性，我们还可以研究解本身的稳定性。

一般来说，稳定解具有以下特征：a) 收敛性：解在特定的条件下趋于一个有限的值。

b) 渐进稳定：解在无穷远处趋于零。

通过稳定性分析，我们可以判断系统是否具有趋于稳定状态的性质，这对于系统控制、优化问题等具有重要意义。

二、微分方程的解存在性分析解的存在性是对微分方程是否能找到满足特定条件的解进行研究。

下面介绍两个常见的解存在性定理。

1. 皮卡-林德勒夫定理对于连续函数f(x,t)和初始条件x(t0)=x0，如果f(x,t)满足利普希茨条件，则方程dx/dt=f(x,t)在区间[t0,t1]上存在唯一的解。

利普希茨条件是指存在一个常数L，使得对于t∈[t0,t1]和x1、x2∈Rn，满足|f(x1,t)-f(x2,t)|≤L|x1-x2|。

2. 广义皮卡-林德勒夫定理对于非线性连续函数f(x)和初始条件x(t0)=x0，如果f(x)满足利普希茨条件，且满足一定的增长条件，则方程dx/dt=f(x)在区间[t0,t1]上存在解。

具有非Lipschitz和非增长条件的带跳倒向随机微分方程秦衍;谢晓敏
【期刊名称】《数学理论与应用》
【年(卷),期】2010(030)003
【摘要】本文研究一类带Poisson跳的倒向随机微分方程.在方程的系数满足非增长条件和非Lipschitz条件下,讨论方程适应解的存在唯一性和稳定性.为了证明解的存在性,首先通过函数变换,构造出一逼近序列,然后运用推广的Bihari不等式和Lebesgue控制收敛定理证明该逼近序列是收敛的,得到逼近序列的极限就是方程的适应解.解的唯一性和稳定性主要运用了Bihari不等式和推广的Bihari不等式来进行证明.
【总页数】9页(P116-124)
【作者】秦衍;谢晓敏
【作者单位】华东理工大学数学系,上海,200237;华东理工大学数学系,上
海,200237
【正文语种】中文
【相关文献】
1.非时齐非Lipschitz条件下带跳的随机发展方程mild解的存在惟一性 [J], 宋玉林;杨翠
2.非Lipschitz条件下的包含下微分算子的带跳倒向随机微分方程 [J], 张孟
3.非Lipschitz条件下带跳的倒向随机微分方程的比较定理 [J], 韩宝燕
4.非Lipschitz条件下带跳倒向随机微分方程解的稳定性 [J], 任永; 夏宁茂
5.非Lipschitz条件下的带跳的倒向随机微分方程 [J], 李娟
因版权原因，仅展示原文概要，查看原文内容请购买。

非Lipschitz条件下多维随机微分方程强解的存在唯一性孙晓君
【期刊名称】《工程数学学报》
【年(卷),期】1998(015)001
【摘要】运用广义Ito公式．在扩散系数非退化非Lipschkz连续，漂流系数可间断的条件下，证明了多维Ito型SDE强解的存在唯一性．
【总页数】4页(P121-124)
【作者】孙晓君
【作者单位】中国纺织大学基础部，上海200051
【正文语种】中文
【中图分类】O211.63
【相关文献】
1.非全局Lipschitz条件下随机延迟微分方程解的存在唯一性定理 [J], 范振成
2.Lipschitz条件下由G-布朗运动驱动的随机泛函微分方程解的存在唯一性 [J], 梁青
3.一类非Lipschitz条件下的量子随机微分方程解的存在唯一性 [J], 让光林
4.Lévy过程驱动的局部非Lipschitz随机微分方程的解的存在唯一性 [J], 苏丽娟
5.随机Lipschitz条件下一般时间终端的多维BSDE解的存在唯一性 [J], 侯杰因版权原因，仅展示原文概要，查看原文内容请购买。

非Lipschitz条件下由连续半鞅驱动的倒向随机微分方程的解李师煜;李文学;李华灿【摘要】经典的倒向随机微分方程以布朗运动为干扰源.研究由连续半鞅驱动的倒向随机微分方程,在生成元满足一定的非Lipschitz条件下,通过构造一个Picard序列的方法,利用It(a)公式、Lebe-sgue控制收敛定理和常微分方程的比较定理,证明其解是存在并且唯一的,对经典倒向随机微分方程进行了实质性的推广.【期刊名称】《黑龙江大学自然科学学报》【年(卷),期】2015(032)004【总页数】5页(P485-489)【关键词】倒向随机微分方程;连续半鞅;非Lipschitz系数;存在性;唯一性【作者】李师煜;李文学;李华灿【作者单位】江西理工大学理学院,赣州341000;江西理工大学理学院,赣州341000;江西理工大学理学院,赣州341000【正文语种】中文【中图分类】O211.6The Backward Stochastic Differential Equations (BSDEs) theory has been a focus of great interest in recent years.To solve a classical BSDE, we look for a couple of processes (y,z) which satisfies the equationwhere T>0 is a finite constant termed the time horizon, ξ is a k-dimensional random variable termed the terminal condition, the randomfunction f:Ω×[0,T]×R×R→Rkis progressively measurable for each(y,z), termed the generator of the BSDE (1), and B is a d-dimensional Brownian motion.The solution (y,z)is a pair of adapted processes.The triple (ξ,T,f) is called the coefficients (parameters) of the BSDE (1).Such equations, in the nonlinear case, were first introduced by Pardoux and Peng[2], who established an existence and uniqueness result for solutions to BSDEs with square integrable parameters under the Lipschitz assumption of the generator f.In particular, existence and uniqueness results of the solutions of BSDE with continuous semi-martingale under Lipschitz condition are obtained, and some properties of the solutions are further discussed by Wang[3].Chen and Wang[4] proposed the following non-uniformly Lipschitz condition for the generator f of multidimensional BSDEs:(H1) dp×dt-a.s.,∀y1,y2∈Rk,z1,z2∈Rk×d,The Euclidean norm of a vector y∈Rk will be defined by |y|, and for a k×d matrix z, we define where z*is the transpose of z.Under (H1), they established the existence and uniqueness of the solution to the BSDE (1) with 0<T<+∞.Fan[1] proposed the following non-Lipschitz assumption for the generator f of multidimensional BSDEs:(H2) dp×dt-a.s.,∀y1,y2∈Rd,z1,z2∈Rk×d,(·)].S[T,a(·),b(·)] denotes the set of functions k(·,·):[0,T]×R+|→ R+satisfying the following two conditions:(1)For fixed t,k(t,·),a continuous, concave and non-decreasing function with k(t,0)=0, and for each t∈[0,T],k(t,u)≤a(t)+b(t)u, where the functions ；(2)The ODE, u′(t)=-k(t,u),t∈[0,T] with u(T)=0, has a unique solutionu(t)=0,t∈[0,T].∞.Under this assumption, Fan[1] proved that the BSDE (1) with 0<T≤+∞ has a unique solution.The other results of existence and uniqueness are obtained by Wang[5], Fan[6], Lepeltier[7], Mao[8] and Li[9-11].Motivated by these results, we are interested in solving one-dimensional BSDEs driven by continuous semi-martingale, where the generators f satisfy (H2) and (H3).We established the existence and uniqueness theorem of solutions for this kind of BSDES.This paper is organized as follows.We introduce some preliminaries and lemmas in Section 1 and put forward and prove our main result in Section 2.Let us first introduce some assumptions and notations, which will be used in this paper.For what follows, let us f ix a number 0<T<+∞.Let(Ω,F,Ρ) be a probability space carrying a standard one-dimensional Brownianmotion(Bt)t≥0, let (Ft)t≥0 be the natural σ-algebra generated by (Bt)t≥0 and F=FT.We shall denote by P, the predictable σ-field.For every positive integer n, we use|·|to denote the norm of a Euclidean space Rn.For t∈[0,T], we define by L2(Ω,FT,Ρ)denote the set of all real-valued, square integral and FT-measurable random process with the property E|ξ|2<+∞.Let<M>be M square variational process.Let S2(0,T;R) be the set of real-valued, adapted and continuous processes (Yt)t∈[0,T] such that ∞.Moreover, for each positive integer n, let M2(0,T;Rd)denote the set of Ft-predictable Rd-valued processes∞.In this paper, we consider the followingone-dimensional backward stochastic differential equationwhere M={Mt,Ft:0≤t<∞}is the continuous semi-martingale with M0=0, and <M>T is bounded, ξ∈L2(Ω,FT,Ρ), yt∈S2(0,T;R),zt∈M2(0,T;Rd).Definition 1 A pair of processes (yt,zt)t∈[0,T] is called a solution to the BSDE(2), if (yt,zt)t∈[0,T]∈S2(0,T;R)×M2(0,T;Rd)and satisfies the BSDE (2). Now, let us introduce the following Lemma 1, which will play an important role in the proof of our main result.Before that, let us first introduce the following assumption on the generator f:(A)dp×dt-a.s.,∀(y,z)∈R×R1×dwhere u(·),v(·):[0,T] both φt and gt are non-negative and Ft-measurable processes with (·)].In order to ensure that the solutions of BSDE(2) are square integrable, we assume d<M>t≤dt,a.s.Using the similar proof of Lemma 1 in Fan[1], one can obtain the following Lemma 1.We omit its proof.Lemma 1 Assume that 0<T<+∞ and f satisfies (A), let (yt,zt)t∈[0,T] be a solution to the BSDE with parameters(ξ,T,f).Then there exist two positive constants c1 and c2such thatholds for each t∈[0,T]In this section, we will show the existence and uniqueness of solution for BSDEs under a weaker condition than the Lipschitz one, which improves the results in Wang[3] and Fan[1].The main result of this paper is the following theorem.Theorem 1 Assume that 0<T<+∞ and that f satisfies (H2) and (H3).Thenfor each ξ∈L2(Ω,FT,Ρ), the BSDE (2) with parameters (ξ,T,f) has a unique solution (yt,zt)t∈[0,T]∈S2(0,T;R)×M2(0,T;Rd).The following lemma will be used in the proof of Theorem 1.Lemma 2 (See Lemma 2 in Fan[1]).Assume that 0<T<+∞andξ∈L2(Ω,FT,Ρ).Assume further that the generator f satisfies (H1) and(H3).Then, the BSDE with parameters (ξ,T,f) has a unique solution.Now, f satisfy (H2) and (H3).We can construct the Picard approximate sequence of the BSDE with para meters (ξ,T,f) as follows:Lemma 3 (See Lemma 3 in Fan[1]).Let 0=T0<T1<T2<…<TN-1<TN, under the hypotheses of Theorem 1, for each t∈[TN-1,TN],n,m≥1, it holds that Lemma 4 (See Lemma 4 in Fan[1]).Under the hypotheses of Theorem 1, there exists a constant K≥0such that for each n≥1 and K.With the help of Lemmas 3 and 4, we can prove Theorem 1.The Proof of Theorem 1Existence Define a sequence of functions {φn(t)}n≥1 as follows:M.Furthermore, by induction one can obtain that for all n≥1, φn(t) satisfies Then, for each t∈[TN-1,TN], the limit of the sequence {φn(t)}n≥1 must exist, we denote it by φ(t).Letting in (5), in view of the facts that ρ(s,·) is a continuous function for each we can deduce from Lebesgue's dominated convergence theorem that for each wheth er T<+∞.Then, by virtue of (H2) one knows that φ(t)=0, t∈[TN-1,T].φ0(s))d<M>s=φ1(t)≤Mthus, by induction one derives that for each m≥1,which means thatis a Cauchy sequence in S2(TN-1,T;R).Furthermore, inview of the facts that ρ(s,·)is a continuous function and ρ(s,0)=0 for eachs∈[0,T],)we also know from (4) and Lebesgue’s dominated convergence theorem that is a Cauchy sequence in M2(TN-1,T;Rd).Define their limits by (yt)t∈[TN-1,T] and (zt)t∈[TN-1,T], respectively.Letting n→∞ in (3) follows that (yt,zt) is a solution to the BSDE with parameters (ξ,T,f) on [TN-1,T].Finally, by replacing TN-1,T and ξ by TN-2,TN-1 and yTN-1respectively in Lemma 1～lemma 4 and the above proof, we can obtain the existence of a solution to the BSDE with parameters (ξ,T,f) on [T N-2,TN-1].Furthermore, repeating the above procedure, one deduce the existence of a solution to the BSDE with parameters (ξ,T,f) on [0,T].Uniqueness Letbe two solutions of the BSDE with parameters (ξ,T,f).).It follows from (H2) thatwhich means that the assumption (A) is satisfied for the generator of the BSDE (6) with μ(t)=α(t),ν(t)=β(t), ψ(·,·)=ρ(·,·),φt≡0 and gt≡0.Then, Lemma 1 yields that for t∈[TN-1,T], T<+∞From the comparison theorem of ODE, we know that where r(t) is the maximum left shift solutio n of the equation: u′(t)=-ρ(t,u);u(T)=0.It follows from (H2) that r(t)=0,t∈[TN-1,T].].].Furthermore, (7) leads that holds true almost surely for t∈[TN-1,T].Thus, we have obtained the uniqueness result on [TN-1,T] and As a result, is also a solution to the BSDE with parameters ).Then we can repeat the above procedure by replacing TN-1 and T by TN-2and TN-1respectively and obtain the uniqueness result on t∈[TN-2,TN-1],and then on the whole [0,T].The proof of Theorem 1 is 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