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

摘要

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 D_t + (1/2)Δ_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, mono-tonicity 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)Δ_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_x 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.