几类带泊松跳随机微分方程数值方法的收敛性与稳定性
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摘要
带有泊松跳的随机微分方程在金融、电子工程、生物等领域具有广泛的应用.由于绝大部分带泊松跳的随机微分方程真解的显式表达式难以获得,所以研究用数值方法求解这类方程具有重要的理论和实际意义.近几年来,关于带泊松跳的随机微分方程,国内外文献仅限于显式或半隐式方法的研究.全隐式方法比显式或半隐式的数值方法具有更好的稳定性.针对几类带泊松跳的随机微分方程,本文研究了全隐式方法的收敛性和稳定性.而且关于带泊松跳的随机微分方程,本文还讨论了1阶强收敛的Milstein方法的收敛性和稳定性.
全文由七章构成.
第一章综述随机微分方程及带泊松跳随机微分方程的理论分析及数值分析的研究概况.
第二章介绍本文需要用到的基础知识,包括概率论、随机过程及随机微分方程等.
第三章研究数值求解带泊松跳随机微分方程的平衡隐式方法的收敛性和均方稳定性.证明了强平衡隐式方法是1/2-阶均方收敛的,并证明了强平衡隐式方法和弱平衡隐式方法当步长充分小时均能保持系统的均方稳定性.
第四章研究数值求解带泊松跳线性随机微分方程的平衡隐式方法的渐近稳定性.证明了在步长充分小的条件下,强平衡隐式方法和弱平衡隐式方法都可以保持系统的渐近稳定性.
第五章建立了关于带泊松跳随机比例微分方程的平衡隐式方法,研究了该方法的均方收敛性与均方稳定性.证明了该方法的强收敛阶为1/2,同时还证明了,对于线性标量方程,当步长充分小时,强平衡隐式方法和弱平衡隐式方法都是均方稳定的.
第六章构造了数值求解带泊松跳中立型随机延迟微分方程的一类隐式单步方法,建立了相容阶和收敛阶之间的关系,获得了一般隐式单步方法的均方收敛阶,并将此结论应用到半隐式方法——随机θ-方法和全隐式方法——平衡隐式方法,获得了这两类方法的收敛阶.
第七章研究数值求解带泊松跳线性随机微分方程的Milstein方法,研究了该方法的均方稳定性和渐近稳定性.证明了强Milstein方法与弱Milstein方法在步长充分小的条件下能保持均方稳定性和渐近稳定性.
数值试验进一步验证了文中所获理论的正确性.
Stochastic differential equations with Poisson-driven jumps arise widely in finance, electrical engineering, biology and so on. In general, it is difficult to obtain the explicit solutions of general stochastic differential equations (SDEs) with jumps. Therefore, solving the SDEs with jumps by the efficient numerical methods is very meaningful in theory and application. In recent years, the researches at home and abroad are only focused on explicit or semi-implicit methods for the SDEs with jumps. Full implicit methods admit better stability property than explicit or semi-implicit methods. This thesis investigates the convergence and the stability of full implicit methods for several classes of SDEs with jumps. Furthermore for the SDEs with jumps, it discusses the convergence and the stability of the Milstein method which has strong convergence rate of one.
This thesis consists of seven parts.
In Chapter1, a survey of modern developments including analytical analysis and numerical analysis for the SDEs and the SDEs with jumps are introduced.
In Chapter2, some elementary concepts including probability theory, stochastic processes, stochastic differential equations et al are presented.
Chapter3studies the convergence and the mean-square stability of the balanced implicit methods for the SDEs with jumps. It is shown that the balanced implicit methods give strong convergence rate of at least1/2. For the linear system, the strong balanced implicit methods and the weak ba-lanced implicit methods are shown to preserve the mean-square stability with the sufficiently small stepsize.
Chapter4investigates the ability of the balanced implicit methods to reproduce the asymptotic stability of the linear SDEs with jumps. It is shown that the asymptotic stability of stochastic jump-diffusion differen-tial equations is inherited by the strong balanced implicit methods and the weak balanced implicit methods with sufficiently small stepsizes.
Chapter5deals with the balanced implicit methods for the stochastic pantograph equations with jumps. The mean-square convergence and the mean-square stability are investigated. It is shown that the balanced imp-licit methods give strong convergence rate of at least1/2. For a linear sca-lar test equation, the strong balanced implicit methods and the weak balan-ced implicit methods are shown to capture the mean-square stability for all sufficiently small time-steps.
In Chapter6, a class of implicit one-step schemes are proposed for the neutral stochastic differential delay equations(NSDDEs) driven by Poisson processes. The relationship between the consistent order and the conver-gence order is established. A general framework for mean-square conver-gence of the methods is provided. The convergence orders of the semi-im plicit schemes——the stochastic θ-methods and the full implicit schemes——the balanced implicit methods are given to illustrate the theoretical results.
In Chapter7, the Milstein method is proposed to approximate the solu-tion of a linear SDEs with jumps. The mean-square stability and the stoch-astically asymptotic stability of the Milstein method are investigated. The strong Milstein method and the weak Milstein method are shown to capture the mean square stability and the asymptotic stability of the system for all sufficiently small time-steps.
The numerical experiments are given to illustrate the theoretical results in the paper.
全文由七章构成.
第一章综述随机微分方程及带泊松跳随机微分方程的理论分析及数值分析的研究概况.
第二章介绍本文需要用到的基础知识,包括概率论、随机过程及随机微分方程等.
第三章研究数值求解带泊松跳随机微分方程的平衡隐式方法的收敛性和均方稳定性.证明了强平衡隐式方法是1/2-阶均方收敛的,并证明了强平衡隐式方法和弱平衡隐式方法当步长充分小时均能保持系统的均方稳定性.
第四章研究数值求解带泊松跳线性随机微分方程的平衡隐式方法的渐近稳定性.证明了在步长充分小的条件下,强平衡隐式方法和弱平衡隐式方法都可以保持系统的渐近稳定性.
第五章建立了关于带泊松跳随机比例微分方程的平衡隐式方法,研究了该方法的均方收敛性与均方稳定性.证明了该方法的强收敛阶为1/2,同时还证明了,对于线性标量方程,当步长充分小时,强平衡隐式方法和弱平衡隐式方法都是均方稳定的.
第六章构造了数值求解带泊松跳中立型随机延迟微分方程的一类隐式单步方法,建立了相容阶和收敛阶之间的关系,获得了一般隐式单步方法的均方收敛阶,并将此结论应用到半隐式方法——随机θ-方法和全隐式方法——平衡隐式方法,获得了这两类方法的收敛阶.
第七章研究数值求解带泊松跳线性随机微分方程的Milstein方法,研究了该方法的均方稳定性和渐近稳定性.证明了强Milstein方法与弱Milstein方法在步长充分小的条件下能保持均方稳定性和渐近稳定性.
数值试验进一步验证了文中所获理论的正确性.
Stochastic differential equations with Poisson-driven jumps arise widely in finance, electrical engineering, biology and so on. In general, it is difficult to obtain the explicit solutions of general stochastic differential equations (SDEs) with jumps. Therefore, solving the SDEs with jumps by the efficient numerical methods is very meaningful in theory and application. In recent years, the researches at home and abroad are only focused on explicit or semi-implicit methods for the SDEs with jumps. Full implicit methods admit better stability property than explicit or semi-implicit methods. This thesis investigates the convergence and the stability of full implicit methods for several classes of SDEs with jumps. Furthermore for the SDEs with jumps, it discusses the convergence and the stability of the Milstein method which has strong convergence rate of one.
This thesis consists of seven parts.
In Chapter1, a survey of modern developments including analytical analysis and numerical analysis for the SDEs and the SDEs with jumps are introduced.
In Chapter2, some elementary concepts including probability theory, stochastic processes, stochastic differential equations et al are presented.
Chapter3studies the convergence and the mean-square stability of the balanced implicit methods for the SDEs with jumps. It is shown that the balanced implicit methods give strong convergence rate of at least1/2. For the linear system, the strong balanced implicit methods and the weak ba-lanced implicit methods are shown to preserve the mean-square stability with the sufficiently small stepsize.
Chapter4investigates the ability of the balanced implicit methods to reproduce the asymptotic stability of the linear SDEs with jumps. It is shown that the asymptotic stability of stochastic jump-diffusion differen-tial equations is inherited by the strong balanced implicit methods and the weak balanced implicit methods with sufficiently small stepsizes.
Chapter5deals with the balanced implicit methods for the stochastic pantograph equations with jumps. The mean-square convergence and the mean-square stability are investigated. It is shown that the balanced imp-licit methods give strong convergence rate of at least1/2. For a linear sca-lar test equation, the strong balanced implicit methods and the weak balan-ced implicit methods are shown to capture the mean-square stability for all sufficiently small time-steps.
In Chapter6, a class of implicit one-step schemes are proposed for the neutral stochastic differential delay equations(NSDDEs) driven by Poisson processes. The relationship between the consistent order and the conver-gence order is established. A general framework for mean-square conver-gence of the methods is provided. The convergence orders of the semi-im plicit schemes——the stochastic θ-methods and the full implicit schemes——the balanced implicit methods are given to illustrate the theoretical results.
In Chapter7, the Milstein method is proposed to approximate the solu-tion of a linear SDEs with jumps. The mean-square stability and the stoch-astically asymptotic stability of the Milstein method are investigated. The strong Milstein method and the weak Milstein method are shown to capture the mean square stability and the asymptotic stability of the system for all sufficiently small time-steps.
The numerical experiments are given to illustrate the theoretical results in the paper.
引文
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