Stability,numerical Calculation And Simulation Of It(?) Stochastic Systems With Jumps

Stochastic systems have become one of the hottest research topics,due to their wide applications in the scientific fields,such as population statistics,biological environment,economy and finance,engineering design,etc.This type of systems takes white Gaussian noise as the only interference sources,to describe a kind of relatively stable and continuous random phenomenon.However,in the real world,the systems may be interfered by some unexpected factors.For example,the sharp stock market oscillation caused by the global financial storm;the extinction of a species caused by the global climate warming,tsunamis,earthquakes and so on.These changes and influence,for the system itself,are stochastic,sudden and unpredictable.This means that the smooth white Gaussian noise alone,which is as the only interference source for the system,can't meet the need of the reality.Therefore,in order to describe the world as real as possible,people introduce the jumping interference,which can depict the abrupt changes.Among the jump systems,one kind is the Markov jumping system.The characteristic of such system is that the dynamic evolution is driven by both the time and the event.The discrete event is the system mode,and the switching of each mode obeys the law of a Markov process.There is also a kind of system named Poisson jumping system,which brings in the Poisson process with continuous time and discrete state,to describe the rapid changes or jumps of some movements in fixed or unfixed time.Stability is one of the essential problems in the system control theory.Stochastic disturbance is often considered as one of the factors making the system unstable.It is an important research direction that,how to study the stability of a system relying on its structure and the conditions of its functions.The stability of a stochastic system includes:the moment exponential stability and almost sure exponential stability,asymptotic stability,stable in probability,T stability,etc.The contents and methods of a stochastic system are richer than that of the ordinary differential system.In addition,due to the characters of nonlinear and coupling in stochastic system,it is very difficult to obtain the analytical solution.So using a discrete numerical method to study the stability of a system is an effective way,which allows us to look inside the internal structure and characteristic of such a system.Numerical methods have been widely studied in economy,biology and neural network models.Therefore,in this work,we focus on the Markov jumping system and the Poisson jumping system,and select the stabilities of the systems and the numerical methods as the main research contents.The main contributions of this thesis are summarized as follows.1.The research background and significance of Markov jumping system and Poisson jumping system are introduced.And then the research progresses and statuses of the two systems and the related neutral stochastic functional system are elaborated emphatically,associated with some relevant theories and methods.2.For the stochastic differential equations with Markov jumps,the almost sure exponential stability of ? numerical method is investigated.Under the conditions that,based on which the trivial solution is almost sure exponentially stable,we prove that the ? method can reproduce the corresponding stability of the trivial solution,by applying the inequality techniques,mathematical expectation,Chebyshev inequality and Borel-Cantelli lemma,combined with the ergodic characteristics of the Markov jumps.This generalizes the results of the Euler-Maruyama method and the Backward Euler-Maruyama method.3.For the nonlinear neutral stochastic delay differential equations with Poisson jumps,the asymptotically mean square stability of the trivial solution is analyzed.And we consider whether the Backward Euler-Maruyama method can reproduce the corresponding stability under the same conditions.When the drift coefficient satisfies a onesided Lipschitz condition,the diffusion and the jumping coefficients satisfy the linear growth condition,we prove that the trivial solution and the numerical solution of the Backward Euler-Maruyama method are asymptotically mean square stable by using the functional comparison principle and the Barbalat lemma.The implicit Backward EulerMaruyama method shows better characteristic than the explicit Euler-Maruyama method for the reason that it works without the linear growth condition on the drift coefficient.4.For the neutral stochastic delay differential equations with Poisson jump,we use another strategy to analyze the exponential mean-square stability of the trivial solution and the numerical solution,which is stronger than the asymptotically mean square stability.The exponential mean-square stability implies the asymptotically mean square stability(the converse is not correct).With some monotone conditions,the trivial solution of the equation is proved to be exponentially mean-square stable by using the stochastic techniques and the properties of Poisson jumps.When the drift coefficient satisfies the linear growth condition,the more general ? method and the split-step ?method are proved to preserve the exponential mean-square stability and the almost sure exponential stability of the trivial solution.5.For the neutral stochastic functional differential equations with Poisson jumps,the almost sure exponential stability and the exponential mean-square stability of the trivial solution are studied.A nonnegative Borel-measurable function is introduced,with its property of?-?0?(?)d? = 1.Under the one-sided Lipschitz condition and the linear growth condition,the Euler-Maruyama method and the Backward Euler-Maruyama method are proved to reproduce the corresponding stability of the trivial solution by using the nonnegative semimartingale convergence theorem.6.For the neutral stochastic differential equations with time-dependent delay and Poisson jumps,we consider the exponential stability of the Backward Euler-Maruyama method.Using the integer part function,stochastic techniques and the nonnegative semimartingale convergence theorem,the numerical solutions produced by the Backward Euler-Maruyama method are proved to be almost sure exponentially stable,then they are also proved to be exponentially mean-square stable.Finally,the full work is summarized and the further research direction is pointed out.The research on the stability and the numerical methods of systems with Markov jumps or Poisson jumps,especially the intensive analysis for different neutral stochastic differential equations with Poisson jumps,not only extends the results of stochastic delay systems and neutral stochastic delay systems to the neutral stochastic delay systems with Poisson jumps,but also enriches the stability theory and numerical methods of stochastic systems with Poisson jumps.