几类随机（偏）微分方程的理论性质与参数估计

This project mainly focuses on the theoretical properties and parameter estimation problems on several typical kinds of stochastic differential equations (SDE) and stochastic partial differential equations (SPDE)..1. As for the specific stochastic heat conduction equations, stochastic Burger equation and stochastic Cahn-Hilliard equation, especially for the related SPDE systems, driven by fractional noise and Levy process, we study the various properties of their solutions, including the existence and uniqueness, the invariant measures, the ergodic properties of the solutions and so on..2. We deeply investigate the theoretical properties of the SDE and SPDE driven by skew Brownian motions, and consider the (reflected) skew Brownian motion, the skew OU process, the skew CIR process and the SPDE just mentioned. To be more specific, we discuss the existence and uniqueness of their solutions, the transition densities, the first hitting time distributions, the stationary distributions and many other properties..3. We explore the statistical inference and parameter estimations of some typical skew diffusion processes, and make further discussion on parameter estimations of several specific SPDEs. By statistical tools or other mathematical methods, we estimate the drift and diffusion coefficients and the skew parameters of the skew diffusion processes. We also concentrate on their economical and financial applications and compare them with the existing models, which may give more insight into the potential applications of skew diffusions.

本项目主要研究几类具体的随机微分方程与随机偏微分方程(及随机偏微分方程组)的理论性质与参数估计。.1， 针对具体的由分式噪声与Levy过程驱动的随机热传导方程，随机Burger方程，随机Cahn-Hilliard 方程等，特别是相关的随机偏微分方程组，我们研究其解的各种理论性质（解的存在唯一性，不变测度，遍历性等）。.2，研究带Skew布朗运动的SDE和SPDE的深入理论性质。其中包括 (带反射的)Skew布朗运动、Skew OU过程、Skew CIR过程等,以及上述提及的SPDE，我们讨论其解的存在唯一性，转移密度，首达时分布，平稳分布等。.3，讨论具体Skew扩散过程的统计推断与参数估计，并进一步考虑一些具体的SPDE的参数估计。通过统计、数学的方法来估计模型的漂移、扩散和Skew扩散系数，考虑其与经济、金融等具体市场模型的比较，体现其应用价值。

该项目主要讨论了分式噪声驱动的随机偏微分方程，包括四阶热方程，带梯度项的随机方程，非高斯Levy过程驱动的随机波动方程等，在理论上讨论解的存在唯一性，长时间行为等。同时，讨论斜扩散过程模型和理论性质，并对具体的Skew CIR模型，提出三叉树网络方法，对基于Skew CIR模型的债券和欧式、美式期权进行仿真数值模拟，并得到比较理想的结果。另外，我们还讨论了一类特殊粘性扩散过程的性质，并给出标的资产价格服从粘性扩散过程时相关欧式期权价格的表达式。同时研究了标的资产服从门限扩散过程且带有跳到违约风险时相关欧式期权的定价问题。