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A Parametric Methodology for $G$-Expectations and Stochastic Systems under Model Uncertainty

Guangqian Zhao

TL;DR

This work tackles stochastic systems under volatility uncertainty by leveraging the G-expectation framework and its dual representation as a supremum over linear expectations. It introduces a parametric homomorphism $\Phi$ that maps the G-expectation space to a family of classical stochastic processes, enabling a linear execution phase for each fixed parameter and a single, uniform synthesis across parameters. The authors prove that $\Phi$ preserves key stochastic structures, including Itô calculus and quadratic variation, and demonstrate the approach on G-SDEs and G-BSDEs, significantly reducing estimation error that accumulates in traditional stepwise sublinear methods. The resulting framework yields sharper, more tractable estimates and provides a unified path for robust pricing and hedging under model uncertainty, with clear avenues for extensions to broader Knightian uncertainties.

Abstract

This paper develops a systematic parametric method for analyzing stochastic systems under volatility uncertainty within the $G$-expectation framework. Leveraging the dual representation of the $G$-expectation as a supremum over a family of probability measures, we operationalize the correspondence between the $G$-expectation space and a parameterized family of classical stochastic processes. Our approach decomposes complex nonlinear analysis into two distinct phases: a ``linear implementation phase,'' which utilizes classical stochastic analysis under a fixed parameter, and a ``consistent estimation phase'' across the parameter space. We rigorously prove that this parametric mapping preserves fundamental stochastic structures, including the Itô integral and quadratic variation, thus enabling a direct transplantation of classical Itô calculus techniques into the $G$-framework. The advantages of this methodology are illustrated through its application to $G$-stochastic differential equations and $G$-backward stochastic differential equations. Furthermore, we demonstrate that this method effectively mitigates the inherent accumulation of estimation errors found in traditional stepwise sublinear procedures, yielding significantly more accurate estimates for fundamental quantities in robust stochastic analysis.

A Parametric Methodology for $G$-Expectations and Stochastic Systems under Model Uncertainty

TL;DR

This work tackles stochastic systems under volatility uncertainty by leveraging the G-expectation framework and its dual representation as a supremum over linear expectations. It introduces a parametric homomorphism that maps the G-expectation space to a family of classical stochastic processes, enabling a linear execution phase for each fixed parameter and a single, uniform synthesis across parameters. The authors prove that preserves key stochastic structures, including Itô calculus and quadratic variation, and demonstrate the approach on G-SDEs and G-BSDEs, significantly reducing estimation error that accumulates in traditional stepwise sublinear methods. The resulting framework yields sharper, more tractable estimates and provides a unified path for robust pricing and hedging under model uncertainty, with clear avenues for extensions to broader Knightian uncertainties.

Abstract

This paper develops a systematic parametric method for analyzing stochastic systems under volatility uncertainty within the -expectation framework. Leveraging the dual representation of the -expectation as a supremum over a family of probability measures, we operationalize the correspondence between the -expectation space and a parameterized family of classical stochastic processes. Our approach decomposes complex nonlinear analysis into two distinct phases: a ``linear implementation phase,'' which utilizes classical stochastic analysis under a fixed parameter, and a ``consistent estimation phase'' across the parameter space. We rigorously prove that this parametric mapping preserves fundamental stochastic structures, including the Itô integral and quadratic variation, thus enabling a direct transplantation of classical Itô calculus techniques into the -framework. The advantages of this methodology are illustrated through its application to -stochastic differential equations and -backward stochastic differential equations. Furthermore, we demonstrate that this method effectively mitigates the inherent accumulation of estimation errors found in traditional stepwise sublinear procedures, yielding significantly more accurate estimates for fundamental quantities in robust stochastic analysis.

Paper Structure

This paper contains 54 sections, 14 theorems, 119 equations, 1 table.

Table of Contents

  1. Introduction
  2. Fundamental Concepts and Parametric Representation
  3. G-Expectation as a Value Function
  4. Dynamic Programming and Nonlinear PDEs
  5. Classical G-Expectation and G-Brownian Motion
  6. G-Normal Distribution
  7. G-Brownian Motion
  8. Parameterized Control Representation
  9. The homomorphism
  10. Notation and Preliminaries
  11. Construction of the homomorphism Phi
  12. Preservation of Algebraic Operations
  13. Analytic Structure Preservation
  14. Itô Calculus Preservation
  15. Itô Calculus homomorphism (Integral Form)
  16. ...and 39 more sections

Key Result

Theorem 2.4

For any cylinder function $\xi = \varphi(B_{t_1}, B_{t_2}-B_{t_1}, \cdots, B_{t_n}-B_{t_{n-1}})$ and $s \leq T$, the conditional G-expectation simplifies to: where for $t_{k-1} \leq s < t_k$, we treat $B_{t_i}$ for $i=1,\dots,k-1$ as constants $x_i$ in the expectation, and for $t \geq s$, we define $B_t^\theta = B_s + \int_s^t \theta_u dW_u$. This definition coincides with the classical G-expecta

Theorems & Definitions (39)

  • Definition 2.1: Construction via Supremum over Linear Expectations
  • Remark 2.1
  • Definition 2.2: Conditional G-Expectation Operator
  • Definition 2.3: Conditional G-Expectation for Cylinder Functions
  • Theorem 2.4: Conditional Expectation Simplification
  • Theorem 2.5: Fundamental Properties
  • Theorem 2.6: Tower Property
  • Lemma 3.1: Density of Cylinder Functions
  • proof
  • Definition 3.2: Uniform $L^p$-Norm
  • ...and 29 more