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G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE
Peng, Shige1,2; Song, Yongsheng3
AbstractBeginning 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.
Keywordbackward SDEs partial differential equations path dependent PDEs G-expectation G-martingale Sobolev space G-Sobolev space
Funding ProjectNSF 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 Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000364983200016
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Document Type期刊论文
Corresponding AuthorPeng, Shige
Affiliation1.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
Recommended Citation
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.
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