带泊松测度随机微分方程数值解的收敛性和稳定性
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摘要
在物理、化学、生物、医学、地质、天文、金融等领域带泊松测度随机微分方程作为数学模型具有重要的理论价值和实际应用意义。该类方程常被用作对一类轨道可以间断的Markov过程建立模型，特别是在金融危机背景下更实际地描述了金融市场的运行行为。由于很难得到带泊松测度随机微分方程精确解的表达式，研究数值解及其性质具有重要的理论和实际意义。本文主要研究了带泊松测度随机微分方程数值解的收敛性和稳定性。
本文首先叙述了带泊松测度随机微分方程的应用背景和研究历史，回顾了其精确解的基本性质，几乎必然稳定性和p (p>2)阶矩稳定性的发展状况。叙述了带泊松测度随机微分方程数值解的研究现状。给出了常用的符号和基本知识。
针对d维带泊松测度随机微分方程，利用泊松测度的补偿测度构造了隐式补偿Euler方法。研究了在全局Lipschitz条件和线性增长条件下隐式补偿Euler方法的均方收敛性。在单边Lipschitz条件下讨论了方程精确解均方指数稳定的条件;研究了在保证方程均方指数稳定的前提下，隐式补偿Euler方法都是均方指数稳定性。最后，给出了数值算例并用Matlab绘图验证了收敛性和稳定性的结论。
针对d维带泊松测度随机微分方程，给出了Euler方法的格式;研究了在非Lipschitz条件下方程的精确解和数值解都以大概率存在于紧集;研究了在非Lipschitz条件下Euler方法的依概率收敛性。给出了数值算例验证得到的收敛性结论。
针对d维带泊松测度随机延迟微分方程，构造了Euler方法，给出了方程全局解的概念并在广义Khasminskii条件下证明了其存在唯一性;研究了在广义Khasminskii条件下方程的精确解和数值解都以大概率存在于紧集,给出了Euler方法的依概率收敛性。对得到的收敛性结论给出了相应的数值算例。
最后，针对d维自变量分段连续型带泊松测度随机微分方程，利用泊松测度的补偿测度构造了隐式补偿Euler方法，定义了方程精确解的概念并证明了在全局Lipschitz条件和线性增长条件下其存在唯一性;研究了在全局Lipschitz条件和线性增长条件下隐式补偿Euler方法的收敛性。在单边Lipschitz条件下讨论了方程均方渐近稳定的条件，研究了在此条件下隐式补偿Euler方法的均方渐近稳定性。
As mathematical models, stochastic differential equations with Poisson measurehave important theoretical value and practical application significance in physics, chem-istry, biology, medicine, geology, astronomy, finance and other fields. This kind of e-quations are often used to establish the models for the discontinuous tracks of Markovprocesses, especially to describe the operation of financial markets in the financial cri-sis. Since it’s difficult to obtain the exact solutions of these equations, the study of thenumerical methods and properties is of great importance in theory and application. Thispaper mainly deals with the convergence and stability of numerical solutions for stochas-tic differential equations with Poisson measure.
This paper first presents the application background and the research history of s-tochastic differential equations with Poisson measure, and then reviews the basic proper-ties of the exact solutions and the development of almost sure stability and p th (p>2)moment stability. The current situation of numerical solutions to stochastic differential e-quations with Poisson measure is also presented, together with commonly used notationsand basic knowledge.
Implicit compensated Euler method is constructed by using the compensated mea-sure of Poisson measure for the d dimensional stochastic differential equations with Pois-son measure. Mean-square convergence of the implicit compensated Euler method isproved under the global Lipschitz conditions and the linear growth conditions. Mean-square exponential stability of the equations is derived under the one-sided Lipschitz con-ditions. On the premise of ensuring the equations to be exponentially stable in meansquare, for arbitrary selected step size, the implicit compensated Euler method is provedto be exponentially stable in mean square. With the numerical examples, the conclusionsof the convergence and stability are verified by Matlab drawing.
The Euler method is given for the d dimensional stochastic differential equationswith Poisson measure. It is proved that the exact solutions and numerical solutions of theequations are in a compact set with a large probability under the non-Lipschitz conditions.The convergence in probability of the Euler method is given under the non-Lipschitzconditions and then is verified by a numerical example.
The Euler method is constructed for the d dimensional stochastic delay differentialequations with Poisson measure. The global solutions for the equations are given andproved to be unique under the generalized Khasminskii-type conditions. It is proved thatthe exact solutions and numerical solutions of the equations are in a compact set with alarge probability under the generalized Khasminskii-type conditions. The convergencein probability of the Euler method is verified. The corresponding numerical example isgiven for the convergence.
Finally, by using the compensated measure of Poisson measure, implicit compensat-ed Euler method is constructed for the d dimensional equations with piecewise continuousarguments driven by Wiener process and Poisson measure. The concept of the exact solu-tions of the equations is defined. The existence and uniqueness of the exact solutions areproved under the global Lipschitz conditions and the linear growth conditions. The con-vergence of the implicit compensated Euler method is verified under the global Lipschitzconditions and the linear growth conditions. Under the one-sided Lipschitz conditons,the mean-square asymptotical stability is given for the equations and it is proved that theimplicit compensaed Euler method is mean-square asymptotically stable.
本文首先叙述了带泊松测度随机微分方程的应用背景和研究历史，回顾了其精确解的基本性质，几乎必然稳定性和p (p>2)阶矩稳定性的发展状况。叙述了带泊松测度随机微分方程数值解的研究现状。给出了常用的符号和基本知识。
针对d维带泊松测度随机微分方程，利用泊松测度的补偿测度构造了隐式补偿Euler方法。研究了在全局Lipschitz条件和线性增长条件下隐式补偿Euler方法的均方收敛性。在单边Lipschitz条件下讨论了方程精确解均方指数稳定的条件;研究了在保证方程均方指数稳定的前提下，隐式补偿Euler方法都是均方指数稳定性。最后，给出了数值算例并用Matlab绘图验证了收敛性和稳定性的结论。
针对d维带泊松测度随机微分方程，给出了Euler方法的格式;研究了在非Lipschitz条件下方程的精确解和数值解都以大概率存在于紧集;研究了在非Lipschitz条件下Euler方法的依概率收敛性。给出了数值算例验证得到的收敛性结论。
针对d维带泊松测度随机延迟微分方程，构造了Euler方法，给出了方程全局解的概念并在广义Khasminskii条件下证明了其存在唯一性;研究了在广义Khasminskii条件下方程的精确解和数值解都以大概率存在于紧集,给出了Euler方法的依概率收敛性。对得到的收敛性结论给出了相应的数值算例。
最后，针对d维自变量分段连续型带泊松测度随机微分方程，利用泊松测度的补偿测度构造了隐式补偿Euler方法，定义了方程精确解的概念并证明了在全局Lipschitz条件和线性增长条件下其存在唯一性;研究了在全局Lipschitz条件和线性增长条件下隐式补偿Euler方法的收敛性。在单边Lipschitz条件下讨论了方程均方渐近稳定的条件，研究了在此条件下隐式补偿Euler方法的均方渐近稳定性。
As mathematical models, stochastic differential equations with Poisson measurehave important theoretical value and practical application significance in physics, chem-istry, biology, medicine, geology, astronomy, finance and other fields. This kind of e-quations are often used to establish the models for the discontinuous tracks of Markovprocesses, especially to describe the operation of financial markets in the financial cri-sis. Since it’s difficult to obtain the exact solutions of these equations, the study of thenumerical methods and properties is of great importance in theory and application. Thispaper mainly deals with the convergence and stability of numerical solutions for stochas-tic differential equations with Poisson measure.
This paper first presents the application background and the research history of s-tochastic differential equations with Poisson measure, and then reviews the basic proper-ties of the exact solutions and the development of almost sure stability and p th (p>2)moment stability. The current situation of numerical solutions to stochastic differential e-quations with Poisson measure is also presented, together with commonly used notationsand basic knowledge.
Implicit compensated Euler method is constructed by using the compensated mea-sure of Poisson measure for the d dimensional stochastic differential equations with Pois-son measure. Mean-square convergence of the implicit compensated Euler method isproved under the global Lipschitz conditions and the linear growth conditions. Mean-square exponential stability of the equations is derived under the one-sided Lipschitz con-ditions. On the premise of ensuring the equations to be exponentially stable in meansquare, for arbitrary selected step size, the implicit compensated Euler method is provedto be exponentially stable in mean square. With the numerical examples, the conclusionsof the convergence and stability are verified by Matlab drawing.
The Euler method is given for the d dimensional stochastic differential equationswith Poisson measure. It is proved that the exact solutions and numerical solutions of theequations are in a compact set with a large probability under the non-Lipschitz conditions.The convergence in probability of the Euler method is given under the non-Lipschitzconditions and then is verified by a numerical example.
The Euler method is constructed for the d dimensional stochastic delay differentialequations with Poisson measure. The global solutions for the equations are given andproved to be unique under the generalized Khasminskii-type conditions. It is proved thatthe exact solutions and numerical solutions of the equations are in a compact set with alarge probability under the generalized Khasminskii-type conditions. The convergencein probability of the Euler method is verified. The corresponding numerical example isgiven for the convergence.
Finally, by using the compensated measure of Poisson measure, implicit compensat-ed Euler method is constructed for the d dimensional equations with piecewise continuousarguments driven by Wiener process and Poisson measure. The concept of the exact solu-tions of the equations is defined. The existence and uniqueness of the exact solutions areproved under the global Lipschitz conditions and the linear growth conditions. The con-vergence of the implicit compensated Euler method is verified under the global Lipschitzconditions and the linear growth conditions. Under the one-sided Lipschitz conditons,the mean-square asymptotical stability is given for the equations and it is proved that theimplicit compensaed Euler method is mean-square asymptotically stable.
引文
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