Research On Dynamical Behavior Of Numerical Methods For Stochastic Systems

The theory of stochastic systems has developed rapidly since the introduction of stochastic integration by Ito in 1951. Whenever applying Lyapunov direct method, Lasalle invariance principle, Razumikhin method or the linear matrix inequality (LMI), the dynam-ical behavior criteria of stochastic systems need to be established through constructing Lya-punov functions or Lyapunov functionals. In addition, due to the complexity of stochastic systems, such systems generally have no analytical solutions. However, with the rapid de-velopment of computer technology, stochastic systems can be simulated by Monte Carlo method rather approximately. When lack of a suitable Lyapunov function or Lyapunov functional, it is also feasible to choose numerical methods and stepsizes to accurately pro-duce the dynamic behavior of analytic solutions, which could provide a new approach for further study of dynamical behavior. Therefore, numerical method has gradually become a quite important research tool for study of stochastic systems.At present, there are plenty of studies on the dynamical behavior of numerical meth-ods of deterministic differential systems. Although some results are based on numerical methods of stochastic systems, still there is a large gap compared with numerical methods of deterministic differential systems. It is mainly because numerical methods of stochastic systems involve dynamical behavior of different numerical methods in various stochastic convergence sense. Thus new techniques and innovation need to be applied to solve those difficulties in the research. Besides, the randomness often appears in financial models. The drift coefficients, diffusion coefficients or jump coefficients always do not satisfy the Lip-schitz condition and the linear growth condition, so the classic methods of dynamical be-havior of stochastic systems are no longer applicable. Therefore, some new methods and inequalities are needed to deal with jump items and coefficients that can not satisfy the Lip-schitz condition or the linear growth condition. Numerical methods as well provide a new access to do research on financial models.The dissertation focuses on the investigation of convergence, convergence order and stability of numerical methods for several stochastic systems. In virtue of Doob martingale inequality, exponential martingale inequality, Borel-Cantelli lemma, stochastic integral in-equalities, together with some algebra inequalities, some research results for Markov jump stochastic systems, neutral stochastic functional systems and Poisson jump stochastic sys-tems are obtained. The main work of this dissertation is as follows:The Split-Step Backward Euler (SSBE) method related to a class of linear stochastic integro-differential systems with Markov jump is established. Then the convergence and convergence order of the SSBE method for the system are discussed. Moreover, using the property of Markov chain, some fundamental inequalities, together with stochastic analy-sis and so on, it gives MS-stability and GMS-stability of the stochastic integro-differential system. Moreover, it presents SSBE method's comparison with the Euler-Maruyama (EM) method and Milstein method, and the SSBE method turns out to be more effective.The EM method of neutral stochastic functional systems is also studied. By stochastic analysis and some fundamental inequalities, the convergence order of the EM method is re-vealed under the global Lipschitz condition. At the same time the pth moment convergence of the EM method of the neutral stochastic functional system is presented under the global Lipschitz condition. Besides, it is shown that under the local Lipschitz condition the con-vergence order in pth moment of the EM method of the neutral stochastic functional system is close to p/2, which is different from the convergence order 1/2 of the EM method of stochastic systems.Taylor method of Poisson jump or Markov jump stochastic systems with delays is dis-cussed by applying Taylor expansion method. Further, some approximation lemmas about the numerical solutions and the step functions are established firstly in virtue of the Doob martingale inequality, the compensate Poisson martingale isometry property and the mean theorem. Then on the basis of these lemmas, the convergence of Taylor method of the system is established.The semi-implicit Euler method of Poisson jump stochastic systems with random mag-nitudes is established. Then by using the discrete Gronwall inequality, the continuous Gron-wall inequality and stochastic analysis theory, the relationship between successive approxi-mation numerical solutions of the system and the step functions is researched. Besides, the convergence of the EM method of the system is given. At the same time the convergence order of the numerical method is revealed as well.The diffusion coefficients of mean-reversing square root process of Poisson jump stochastic systems do not satisfy both the Lipschitz condition and the linear growth condi-tion so a number of new techniques are developed to overcome the difficulty. Through con-structing new finite sequences, this dissertation discusses the boundedness of non-negative solutions of the system, the mean reversion of the numerical method of the system, then the proof of the convergence of the numerical method, and finally the application of the numer-ical method in price calculation of some financial products such as bonds and options.The diffusion coefficients of mean-reversingγ-process of Poisson jump stochastic sys-tems do not satisfy the linear growth condition so new techniques are applied to solve the difficulty. Various boundedness of the positive solutions and the mean reversion of the nu- merical solutions are discussed before hand. In addition, the convergence in probability of the numerical method is proved and the application of numerical method in price calculation of the financial products like bonds and options are introduced.The study on dynamical behavior of numerical methods for Markov jump stochastic integro-differential systems, neutral stochastic functional systems and Poisson jump stochastic systems in this dissertation deeply reveals dynamical mechanism of numerical methods of stochastic systems, which not only enriches the numerical analysis theory, but also extends the approach for the research on the dynamical behavior of stochastic systems. Numerical examples also illustrate the validity of the results and the effectiveness of the proposed methods.