[Paper Review] G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE

Shigē Péng, Yongsheng Song|arXiv (Cornell University)|May 21, 2013

Stochastic processes and financial applications24 references4 citations

TL;DR

This paper introduces G-expectation weighted Sobolev spaces (G-Sobolev spaces) and establishes a one-to-one correspondence between a class of backward SDEs driven by G-Brownian motion and path-dependent PDEs in these spaces. The key contribution is a rigorous link between G-BSDEs and fully nonlinear path-dependent PDEs, extending classical BSDE-PDE duality to sublinear expectations and non-Markovian settings.

ABSTRACT

We introduce a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian motion are in fact path dependent PDEs in the corresponding Sobolev spaces under G-norms. For the linear case of G corresponding the classical Wiener probability space with Wiener measure P, we have established a 1-1 correspondence between BSDE and such new type of quasilinear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness of such type of fully nonlinear path-dependence PDE in G-Sobolev space have been obtained via a recent results of BSDE driven by G-Brownian motion.

Motivation & Objective

• To develop a new class of Sobolev-type function spaces weighted by G-expectation, termed G-Sobolev spaces, to handle path-dependent stochastic dynamics.
• To establish a rigorous duality between backward SDEs driven by G-Brownian motion and fully nonlinear path-dependent PDEs.
• To extend the classical BSDE-PDE correspondence to non-Markovian, sublinear expectation frameworks using G-martingale theory.
• To formulate weak solutions in an expanded G-Sobolev space $ W_{\mathcal{A}_{G}}^{\frac{1}{2},1;p}(0,T) $, allowing for weaker derivatives and broader applicability.
• To recover the classical linear case as a special case when $ G(a) = \frac{a}{2} $, linking to Wiener measure and standard PDEs.

Proposed method

• Define G-Sobolev spaces $ W_{G}^{1,2;p}(0,T) $ as completions of smooth cylinder functions under norms involving G-expectation and derivatives of path-dependent functions.
• Introduce a backward SDE of the form $ Y_t = \xi + \int_t^T f(s,Y_s,Z_s,\eta_s)ds - \int_t^T Z_s dB_s - (K_T - K_t) $, where $ K_t = \frac{1}{2}\int_0^t \eta_s d\langle B\rangle_s - \int_0^t G(\eta_s)ds $, modeling non-Markovian dynamics under model uncertainty.
• Establish a 1-1 correspondence between solutions of this G-BSDE and viscosity solutions of the path-dependent PDE $ D_t u + G(D_x^2 u) + f(t,u,D_x u,D_x^2 u) = 0 $ with terminal condition $ u(T,\omega) = \xi(\omega) $.
• Introduce an expanded weak solution space $ W_{\mathcal{A}_{G}}^{\frac{1}{2},1;p}(0,T) $ to accommodate solutions with weaker regularity, particularly for the case when $ f $ is independent of $ D_x^2 u $.
• Use G-martingale decomposition and approximation techniques (e.g., time-averaging via $ \eta_t^h $) to prove convergence and uniqueness in the G-framework.
• Leverage the G-Itô formula and properties of sublinear expectations to ensure consistency and stability of solutions under $ L_G^p $-norms.

Experimental results

Research questions

• RQ1 How can Sobolev-type spaces be generalized to accommodate path-dependent functionals under sublinear expectations?
• RQ2 What is the precise correspondence between backward SDEs driven by G-Brownian motion and path-dependent PDEs?
• RQ3 Can weak solutions be formulated in a broader function space to include less regular solutions, particularly in the linear or degenerate case?
• RQ4 How does the G-BSDE framework recover the classical linear BSDE-PDE duality when $ G(a) = \frac{a}{2} $?
• RQ5 What are the necessary and sufficient conditions for existence and uniqueness of solutions in the G-Sobolev space framework?

Key findings

• A one-to-one correspondence is established between solutions of G-BSDEs and viscosity solutions of path-dependent PDEs in the G-Sobolev space $ W_G^{1,2;p}(0,T) $.
• For the case where $ f $ is independent of $ D_x^2 u $, a weak solution formulation in $ W_{\mathcal{A}_G}^{\frac{1}{2},1;p}(0,T) $ is introduced, which generalizes $ W_G^{1,2;p}(0,T) $ and matches the G-BSDEs studied in [HJPS12].
• The solution space $ W_G^{1,2;p}(0,T) $ is shown to be complete under the G-expectation-weighted norm, ensuring well-posedness of the PDEs.
• When $ G(a) = \frac{a}{2} $, the framework reduces to the classical Wiener probability space, and the correspondence recovers the standard BSDE-PDE duality with solutions in a Sobolev space weighted by Wiener measure.
• The paper proves that if a sequence $ \{\eta_n\} $ is Cauchy in the $ \tilde{M}_G^p $-norm and $ \|\eta_n\|_{M_G^p} \to 0 $, then $ \|\eta_n\|_{\tilde{M}_G^p} \to 0 $, ensuring convergence in the extended space.
• Using G-martingale decomposition and time-averaging approximations, the paper proves that $ \int_0^t \eta_s^+ ds = 0 $ and $ \int_0^t \eta_s^- ds = 0 $, implying $ \|\eta\|_{\tilde{M}_G^p} = 0 $, which confirms uniqueness and stability of solutions.