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G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE

文献类型：期刊论文

作者Peng, Shige1,2; Song, Yongsheng3
刊名JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN
出版日期2015-10-01
卷号67期号:4页码:1725-1757
ISSN号0025-5645
关键词backward SDEs partial differential equations path dependent PDEs G-expectation G-martingale Sobolev space G-Sobolev space
DOI10.2969/jmsj/06741725
英文摘要Beginning from a space of smooth, cylindrical and nonanticipative processes defined on a Wiener probability space (Omega, F, P), we introduce a P-weighted Sobolev space, or "P-Sobolev space", of non-anticipative path-dependent processes u = u(t, omega) such that the corresponding Sobolev derivatives D-t + (1/2)Delta(x) and D(x)u of Dupire's type are well defined on this space. We identify each element of this Sobolev space with the one in the space of classical L-P(p) integrable Ito's process. Consequently, a new path-dependent Ito's formula is applied to all such Ito processes. It follows that the path-dependent nonlinear Feynman-Kac formula is satisfied for most L-P(p)-solutions of backward SDEs: each solution of such BSDE is identified with the solution of the corresponding quasi-linear path-dependent PDE (PPDE). Rich and important results of existence, uniqueness, monotonicity and regularity of BSDEs, obtained in the past decades can be directly applied to obtain their corresponding properties in the new fields of PPDEs. In the above framework of P-Sobolev space based on the Wiener probability measure P, only the derivatives D-t + (1/2)Delta(x) and D(x)u are well-defined and well-integrated. This prevents us from formulating and solving a fully nonlinear PPDE. We then replace the linear Wiener expectation E-P by a sub-linear G-expectation E-G and thus introduce the corresponding G-expectation weighted Sobolev space, or "G-Sobolev space", in which the derivatives D(t)u, D-x(u) and D(x)(2)u are all well defined separately. We then formulate a type of fully nonlinear PPDEs in the G-Sobolev space and then identify them to a type of backward SDEs driven by G-Brownian motion.
资助项目NSF of China[10921101] ; 111 Project[B12023] ; NCMIS ; Youth Grant of National Science Foundation[11101406] ; Key Lab of Random Complex Structures and Data Science, CAS[2008DP173182]
WOS研究方向Mathematics
语种英语
出版者MATH SOC JAPAN
WOS记录号WOS:000364983200016
源URL[http://ir.amss.ac.cn/handle/2S8OKBNM/21318]  
专题应用数学研究所
通讯作者Peng, Shige
作者单位1.Shandong Univ, Sch Math, Jinan, Peoples R China
2.Shandong Univ, Qilu Inst Finance, Jinan, Peoples R China
3.Chinese Acad Sci, Acad Math & Syst Sci, Beijing, Peoples R China
推荐引用方式
GB/T 7714		 Peng, Shige,Song, Yongsheng. G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE[J]. JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN,2015,67(4):1725-1757.
APA Peng, Shige,&Song, Yongsheng.(2015).G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE.JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN,67(4),1725-1757.
MLA Peng, Shige,et al."G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE".JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 67.4(2015):1725-1757.

入库方式： OAI收割

来源：数学与系统科学研究院

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