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

doi:10.2969/jmsj/06741725

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	G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE
	Peng, Shige 1,2; Song, Yongsheng3
刊名	JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN
	2015-10-01
卷号	67 期号:4 页码:1725-1757
关键词	backward SDEs partial differential equations path dependent PDEs G-expectation G-martingale Sobolev space G-Sobolev space
ISSN号	0025-5645
DOI	10.2969/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
内容类型	期刊论文
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.