随机过程论中若干问题的研究

In this project, the following problems have been researched. (1) The limit theorems of vector-valued semimartingales were given. In this part, a new class of stochastic processes (Φ’-valued martingale measures) was introduced and the limit.theorems of Hilbert-valued semimartingales and Φ’-valued martingale measures and the stability of stochastic differentical equations on Hilbert space and on Φ’ were.studied. (2) Stochastic differentical equations with jumps were researched. In this part, we gave the stability, the existence of Lyapunov exponent and invariant measures and approximation of Lyapunov exponent and invariant measure of stochastic differentical equations with jumps. Random dynamical systems with jumps were also stuied in this.part. (3) Integration of integrable systems with stochastic perturbation terms were stuied. We gave the integration of stochastic KdV equation, stochastic mKdV equation and stochastic Burgers equation. (4) Branching models in random environments were researched. In this part, we mainly studied PSD branching models. We gave the conditions for the existence of stationary distribution and the calculation of stationary distribution. Moreover, we also give some applications of branching.models in random network. (6) We studied the Morkov selection and representation of.set-valued Markov processes, the regularity of the paths of set-valued (super, sub) martingales with continuous time. Moreover, the Doob-Meyer decomposition theorem.and the localization of set-valued supermartingales were research in this project. In this project, we have had that 10 papers have been published (there are 2 SCI in them) , 7 papers have been accepted (there are 4 SCI in them) and 17 papers are being gone over manuscripts.

本项目主要包括：随机过程极限定理、向量值半鞅理论以及随机分析在金融经济学中应用的研究：平稳环境、马氏环境中的PSD分枝过程的研究、马氏环境中反射布朗运动及相关随机纭⒁毫髂Ｐ偷难芯浚航⒓德硎瞎毯徒调钡囊话憷砺邸Ｕ庑┪侍舛际撬婊搪壑械娜鹊愫湍训阄侍猓云渖钊胙芯吭诶砺凵虾陀τ蒙隙季哂兄匾庖濉