G-布朗驱动下的非线性中立型随机延迟微分方程解的存在唯一性研究  

The existence-uniqueness of solutions for nonlinear neutralstochastic delay differential equations driven by G-Brownian motion

作　　者：张可为[1] 袁海燕[1] ZHANG Kewei;YUAN Haiyan(College of Science,Heilongjiang Institute of Technology,Harbin 150050,China)

机构地区：[1]黑龙江工程学院理学院,哈尔滨150050

出　　处：《黑龙江工程学院学报》2022年第2期1-6,19,共7页Journal of Heilongjiang Institute of Technology

基　　金：国家自然科学基金青年基金项目(11901173);黑龙江省自然科学基金联合引导项目(LH2019A030);黑龙江工程学院省级领军人才梯队培育计划(2020LJ01);黑龙江省属本科高校基本科研业务费科研项目(2021GJ08)。

摘　　要：随机微分方程解析解显式表达很难获得,数值解及其相关性质的研究成为关注热点。解析解的存在及唯一性是进行数值计算的前提。虽然关于线性随机微分方程及非线性随机微分方程解的存在唯一性及有界性相关研究结论已经很丰富了,但是关于依赖于过去状态变化的G-布朗驱动下的中立型随机延迟微分方程解的研究却尚未发现。文中首先将G-布朗驱动下的中立型随机延迟微分方程等价为积分微分方程,利用矩阵范数的定义及Holder不等式、Gronwall不等式、BDG不等式及Cp不等式的性质给出G-布朗运动驱动下的非线性中立型随机延迟微分方程解析解的有界估计。之后通过定义Picard迭代格式,利用文献[8]中的推论8及Doob鞅不等式、Chebyshev不等式及Borel-Cantelli引理证明了G-布朗运动驱动下的非线性中立型随机延迟微分方程解析解的存在性。Since most stochastic differential equations(SDEs)cannot be solved explicitly,numerical approximations which are on the basis of incorporating the stochastic factor in the classical numerical approximations for SDEs have become an important tool in the study of SDEs.The existence-uniqueness theory for solutions of these equations are significant basic conditions and have received a huge attention.Although the results on the existence and uniqueness of solutions for linear and nonlinear stochastic differential equations are very rich,up to the best of our knowledge,little seems to be known about the existence and uniqueness for solutions of G-NNSDDEs,the aim of this paper is to fill the gap.Firstly,this paper equivalent the neutral stochastic delay differential equation to an integro differential equation.Then it applies the definition of matrix norm and the properties of Holder inequality and Gronwall inequality,BDG inequality and Cp inequality to giving the bounded estimates of the analytic solutions of nonlinear neutral stochastic delay differential equations driven by G-brownian motion.Finally,this paper defines the Picard approximation scheme and use the proposition 8 in[8],Doob inequality,Chebyshev inequality and Borel-Cantelli lemma to establish the uniqueness and existence theorem.

关 键 词：非线性中立型随机延迟微分方程 G-布朗运动 Picard迭代 ITO公式 

分 类 号：O211.63[理学—概率论与数理统计]