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

• Peng Shige

  School of Mathematics and Qilu Institute of Finance, Shandong University

• Song Yongsheng

  Academy of Mathematics and Systems Science

書誌事項

タイトル別名

• <i>G</i>-expectation weighted Sobolev spaces, backward SDE and path dependent PDE

抄録

Beginning from a space of smooth, cylindrical and non-anticipative processes defined on a Wiener probability space (Ω, ℱ, P), we introduce a P-weighted Sobolev space, or “P-Sobolev space”, of non-anticipative path-dependent processes u = u(t,ω) such that the corresponding Sobolev derivatives \mathcal{D}_t + (1/2)Δ_x and \mathcal{D}_xu 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 Itô's process. Consequently, a new path-dependent Itô's formula is applied to all such Itô processes. <br> 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. <br> In the above framework of P-Sobolev space based on the Wiener probability measure P, only the derivatives \mathcal{D}_t + (1/2)Δ_x and \mathcal{D}_xu 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 sublinear G-expectation \mathbb{E}^G and thus introduce the corresponding G-expectation weighted Sobolev space, or “G-Sobolev space”, in which the derivatives \mathcal{D}_tu, \mathcal{D}_xu and \mathcal{D}_x²u 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.

収録刊行物

• Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ

  Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ 67 (4), 1725-1757, 2015

  一般社団法人 日本数学会

キーワード

• backward SDEs
• partial differential equations
• path dependent PDEs
• <i>G</i>-expectation
• <i>G</i>-martingale
• Sobolev space
• <i>G</i>-Sobolev space

詳細情報

• CRID: 1390282680092040064
• NII論文ID: 130005108959
• NII書誌ID: AA0070177X
• DOI: 10.2969/jmsj/06741725
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• NDL書誌ID: 026817771
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I026817771
• 本文言語コード: en
• データソース種別:
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