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Pathwise Patching: A Geometric Construction of Quasi-Sure Solutions to G-SDEs

Guangqian Zhao

TL;DR

This work develops a geometric, pathwise framework for constructing quasi-sure solutions to G-SDEs under model uncertainty by patching measure-specific solutions on the path space. Pathwise uniqueness guarantees that X^P and X^Q agree on overlaps, enabling a universal, quasi-sure process X that solves the GSDE and is universally measurable. It further introduces a robust Malliavin calculus based on variational equations, bypassing measure-dependent $L^2$-closures and ensuring derivatives align with classical Malliavin derivatives across all models in the family. The approach provides a transparent, path-space interpretation of quasi-sure analysis with potential implications for robust finance, stochastic control, and numerical methods under Knightian uncertainty.

Abstract

This paper explores a geometric approach to constructing quasi-sure solutions for $G$-stochastic differential equations (G-SDEs) under model uncertainty. We propose a pathwise patching methodology that systematically combines measure-specific solutions into a unified, universally measurable process. The construction relies on pathwise uniqueness and the convex structure of the $G$-expectation framework to ensure compatibility across different probability measures. We further investigate the possibility of developing a robust Malliavin calculus within this framework. By reformulating the Malliavin derivative through variational equations that inherit the quasi-sure structure, we attempt to overcome the challenges posed by traditional $L^2$-closure arguments in multi-measure settings. While our approach offers enhanced geometric intuition and a potentially more transparent foundation for stochastic analysis under model uncertainty, we recognize its limitations and the need for further investigation into its full scope and applicability. The results may have implications for robust financial mathematics and stochastic control, though we present them as a preliminary step toward more comprehensive theories.

Pathwise Patching: A Geometric Construction of Quasi-Sure Solutions to G-SDEs

TL;DR

This work develops a geometric, pathwise framework for constructing quasi-sure solutions to G-SDEs under model uncertainty by patching measure-specific solutions on the path space. Pathwise uniqueness guarantees that X^P and X^Q agree on overlaps, enabling a universal, quasi-sure process X that solves the GSDE and is universally measurable. It further introduces a robust Malliavin calculus based on variational equations, bypassing measure-dependent -closures and ensuring derivatives align with classical Malliavin derivatives across all models in the family. The approach provides a transparent, path-space interpretation of quasi-sure analysis with potential implications for robust finance, stochastic control, and numerical methods under Knightian uncertainty.

Abstract

This paper explores a geometric approach to constructing quasi-sure solutions for -stochastic differential equations (G-SDEs) under model uncertainty. We propose a pathwise patching methodology that systematically combines measure-specific solutions into a unified, universally measurable process. The construction relies on pathwise uniqueness and the convex structure of the -expectation framework to ensure compatibility across different probability measures. We further investigate the possibility of developing a robust Malliavin calculus within this framework. By reformulating the Malliavin derivative through variational equations that inherit the quasi-sure structure, we attempt to overcome the challenges posed by traditional -closure arguments in multi-measure settings. While our approach offers enhanced geometric intuition and a potentially more transparent foundation for stochastic analysis under model uncertainty, we recognize its limitations and the need for further investigation into its full scope and applicability. The results may have implications for robust financial mathematics and stochastic control, though we present them as a preliminary step toward more comprehensive theories.

Paper Structure

This paper contains 17 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Literature Review and Theoretical Context
  3. Foundations of G-Expectation and Stochastic Analysis Under Uncertainty
  4. Pathspace Methods and Rough Path Theory
  5. Connections with Classical Stochastic Analysis and Applications
  6. Future Research Directions
  7. Preliminaries
  8. The Two-Measure Case
  9. The Universal Solution via Pathwise Patching
  10. The Geometric Construction
  11. Fundamental Properties
  12. Universal Measurability
  13. Robust Malliavin Calculus via Variational Equations
  14. The Classical Malliavin Derivative and Its Limitations
  15. Robust Malliavin Derivative for Stochastic Integrals
  16. ...and 2 more sections

Key Result

Lemma 4.1

zhang2017 Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathscr{P}$ be a family of probability measures on $(\Omega, \mathcal{F})$. Let $\mathbb{F} = (\mathcal{F}_t)_{t \in [0,T]}$ be a filtration satisfying the usual conditions under each $\mathbb{P} \in \mathscr{P}$.The canonical pr Then, there exists a universally measurable process $M$ (with respect to $\mathscr{P}$) such that f

Theorems & Definitions (29)

  • Definition 3.1: Quasi-sure property
  • Definition 3.2: Universally Measurable Sigma-Algebra
  • Definition 3.3: Universally Measurable Function
  • Remark 3.1: The Role of Universal Measurability
  • Definition 3.4: G-Stochastic Differential Equation
  • Definition 3.5: Sublinear Expectation and G-Expectation
  • Remark 3.2
  • Remark 4.1
  • Lemma 4.1: Quasi-Sure Robustness of Itô Integration
  • proof : Proof Sketch
  • ...and 19 more