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Well-posedness of high dimensional degenerate SDEs

Abstract

We consider the SDE degenerated at the boundary of a non-smooth domain. We prove uniqueness and existence of the martingale problem related to this degenerate SDEs under suitable non-negativity and regularity conditions on the coefficients. Applying martingale problem theory of Stroock and Varadhan, we turn the uniqueness problem of the SDE to the well-posedness of a kind of degenerate PDE with Neumann boundary condition defined on $R_{+}^{n}$. The difficulties for solvability of the problem mainly come from the degeneration of the operator, domain with corner, and correlation of different components of the SDE. The Schauder estimate for the degenerate PDE is given. We first estimate the mixed second order derivatives, and then utilize the perturbation method and the Schauder estimation for diagonal form to deal with the second order derivatives of normal direction on the corner boundary. This is a joint work with Kai Du.


报告人简介:张伏,男,讲师,现在上海理工大学理学院工作。2004年毕业于中国矿业大学理学院,获理学学士;2009年于南京大学数学系获理学硕士学位;2013年于复旦大学获理学博士学位,研究方向为随机控制。后在复旦大学管理学院从事金融数学方向博士后研究。现在研究方向为随机控制、随机分析与偏微分方程。主持国家自然科学基金面上项目1项,青年基金项目1项。多篇文章在《SIAM J. Contrl. Optimal》、《Stochastic Process. Appl.》、《Ann. Inst. Henri Poincaré Probab. Stat》、《Discrete Contin. Dyn. Syst.》等学术杂志发表。