超过程和连续状态分枝过程(CB 过程)是两类最主要的连续时空分枝过程，它们分别作为测度值和实值过程在物理、生物和随机金融等学科中有着重要的应用价值，随机方程是研究连续时空分枝过程最重要的工具之一。现实的随机扰动中经常出现带跳现象，这使得研究带跳随机方程具有重大的现实意义和迫切的社会需求。同时，带跳随机方程的研究也为连续时空分枝过程的研究注入了新的活力。. 本项目在理论方面，拟在现有研究基础上，完善超过程理论与相关带跳且非 Lipschitz 系数随机偏微分方程(SPDE)解的性质(轨道唯一性等)，探索与发展超过程和 SPDE 的新理论和新方法，研究带竞争的连续状态非线性分枝过程及相关带跳随机微分方程。在应用方面，拟研究一类 CB 过程满足的金融模型(带跳 CIR 模型)和其形式上相似的分数 Lévy 过程驱动的 CIR 模型的参数估计。

Superprocesses and continuous-state branching processes (CB processes) are two of the main continuous space-time branching processes. As measure-valued and real-valued processes, respectively, they find important applications in the fields of physics, biology and stochastic finance. One of the most important techniques in the study of continuous space-time branching processes is the associated stochastic equations. People often introduce jumps to describe phenomenon with drastic fluctuations over short time periods to find more realistic modeling of random disturbance. Hence, the study of stochastic equations with jumps is both of practical significance and of urgent demands by the society. At the same time, the study of stochastic equations with jumps can also bring new vitality to the research of continuous space-time branching processes. . On the theoretical side, based on existing research we will improve the theories of superprocesses and the properties of solutions to the related stochastic partial differential equations (SPDE) with jumps and non-Lipschitz coefficient (e.g. pathwise uniqueness) in this project. Then new theories and new methods of superprocesses and SPDEs with jumps will be explored and further developed. We will also study continuous-state non-linear branching processes with competition and their related stochastic differential equations. On the application side, we will study the parameter estimation of a CB process, which is also a financial model (CIR model with jumps), and a CIR model driven by a fractional Lévy process, which has a similar form as the CIR model with jumps.