布朗运动和双分式布朗运动驱动的非Lipschitz随机微分方程的一些结论

The Malliavin calculus is an infinite-dimensional differential calculus on the Wiener space. This theory has became an important branch of the theory of stochastic analysis . In this thesis, we mainly study three aspects, one hand is some applications of Malliavin calculus, on the other hand, some problems of multivalued stochastic differential equations, Finally, some problems of backward stochastic differential equations. The detailed content of the thesis are as follows：.(1) We will give the results about the fractional smoothness of derivative of self-intersection local times with respect to Brownian motion and fractional Brownian motion, Compare with the existing works, based on K-method, Firstly, we will improve the result of p=2 to any p>1; Secondly,a is fractional；.(2) We will establish various limit theorems for one-dimensional stochastic variational inequalities with Yamada-Watanabe type conditions on the coefficients, including, e.g.,the construction of the solution by Euler scheme, the convergence of the Yosida approximation and stability of the solution. Besides, convergence rates will be presented for these two approximations when the coefficients are only Holder continuous；.(3) Finally, we will study the Malliavin's differentiability of BSDEs driven by BM and Poisson random measure with non-Lipschitz generator. For the existence and uniqueness of solution in the case of backward stochastic differential equations, previous papers is Lipschitz, but in our work, we will allow that is non-Lipschitz, the difficulty lies in that we can not use fixed point theorem, thus we can not directly use iterations. After obtain the theorem about the existence and uniqueness of solution in the case of backward stochastic differential equations, we will also use it to get the Malliavin differentiability of the solution of general backward stochastic differential equation.

Malliavin分析是一种高斯概率空间上的无穷维随机分析，该理论已经成为随机分析理论的一个重要分支. 本项目将重点研究三方面的内容, 一是推导Malliavin计算的若干应用, 二是优化随机变分方程的若干极限定理问题，三是研究倒向随机微分方程的解的性质.具体内容如下：.(1)本项目拟推导出关于布朗运动及分式布朗运动局部时的光滑性的结果.与已有的文献相比,基于k方法,预期将p=2的结果改进为对任意的p>1成立;其次a是分数次的；.(2)对一维的,带Yamada-Watanabe型系数条件的随机变分方程,本项目拟建立不同条件下的极限定理,包括通过欧拉折线逼近的解的构造,Yamada逼近的收敛性和解的稳定性.且当系数只是Holder连续时,拟给出两种逼近方法的收敛率；.(3)最后本项目拟研究由布朗运动和泊松随机测度诱导的倒向随机微分方程的解的Malliavin可微性.

在第一个成果中，令B^(H_i,K_i )={B_t^(H_i,K_i ),t≥0 }（i=1,2）是两个独立的双分式布朗运动，参数分别为H_i∈(0,1),K_i∈(0,1]。在这篇论文中，我们考虑双分式布朗运动的局部时和自相交局部时的导数的分式光滑性，以及两个独立双分式布朗运动的相交局部时的导数的光滑性。在此我们用的方法是K方法。在第二个成果中，我们考虑由分式布朗运动驱动的一维广义均场倒向随机微分方程，即这类方程的系数不仅依赖解本身而且依赖解的分布。我们首先给出系数在Lipschitz条件下，这类均场倒向随机微分方程的比较定理，然后我们证明当系数只有连续性和线性增长时，均场倒向随机微分方程解的存在性。唯一性仍是个开问题。