[Paper Review] On the Representation Theorem of G-Expectations and Paths of G--Brownian Motion

Mingshang Hu, Shigē Péng|ArXiv.org|Apr 29, 2009

Stochastic processes and financial applications8 references4 citations

TL;DR

This paper provides a simplified, elementary proof of the representation theorem for G-expectations using only basic probability theory and the Hahn-Banach theorem, avoiding complex stochastic control methods. It establishes that bounded continuous functions on continuous path space are in the G-expectation completion of Lipschitz cylinder functions, enabling foundational results in G-Brownian motion and stochastic calculus.

ABSTRACT

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures $\{P_θ:θ\in Θ\}$ to represent an important sublinear expectation--G-expectation $\mathbb{E}[\cdot]$. We also give a concrete approximation of a bounded continuous function $X(ω)$ by an increasing sequence of cylinder functions $L_{ip}(Ω)$ in order to prove that $C_{b}(Ω)$ belongs to the $\mathbb{E}[|\cdot|]$-completion of the $L_{ip}(Ω)$.

Motivation & Objective

• To provide a direct, elementary proof of the representation theorem for G-expectations without relying on advanced stochastic control theory.
• To construct a weakly compact family of probability measures that represent a G-expectation using only basic probabilistic tools.
• To prove that every bounded continuous function on the space of continuous paths belongs to the G-expectation completion of the space of Lipschitz cylinder functions.
• To establish the foundational completeness result necessary for the development of stochastic calculus under model uncertainty.

Proposed method

• Uses the Hahn-Banach theorem to represent sublinear expectations as the supremum over a family of linear expectations.
• Constructs a weakly compact family of probability measures {P_θ : θ ∈ Θ} that represent the G-expectation via elementary approximation techniques.
• Introduces a concrete increasing sequence of Lipschitz cylinder functions L_ip(Ω) that approximate any bounded continuous function X(ω) on the path space Ω.
• Employs piecewise linear interpolation on time grids to define approximating functions ω^(n)(ω), ensuring uniform convergence on compact subsets.
• Applies the G-expectation norm to control approximation error, leveraging compactness and continuity of X.
• Combines approximation error bounds with measure-theoretic estimates to show convergence in the G-expectation L^1 norm.

Experimental results

Research questions

• RQ1 Can the representation of G-expectations be proven without invoking sophisticated stochastic control theory?
• RQ2 Is there a direct construction of a weakly compact family of probability measures that represent a G-expectation using only elementary probability tools?
• RQ3 Can every bounded continuous function on the space of continuous paths be approximated in the G-expectation L^1 norm by Lipschitz cylinder functions?
• RQ4 What is the relationship between the space of bounded continuous functions and the G-expectation completion of L_ip(Ω)?
• RQ5 How can the G-Brownian motion framework be grounded on a minimal set of analytical assumptions?

Key findings

• The paper provides a new, elementary proof of the representation theorem for G-expectations using only the Hahn-Banach theorem and basic probability theory.
• A weakly compact family of probability measures {P_θ : θ ∈ Θ} is explicitly constructed to represent the G-expectation as a supremum over expectations under these measures.
• Every bounded continuous function X(ω) on the space of continuous paths is shown to be in the G-expectation L^1-completion of the space of Lipschitz cylinder functions L_ip(Ω).
• The approximation of X(ω) by an increasing sequence of Lipschitz cylinder functions is constructed explicitly via time-discretized piecewise linear interpolation.
• The result implies that L_G^1(Ω) = L_c^1 and L_G^p(Ω) = L_c^p for all p > 0, establishing the completeness of the space under G-expectation.
• The framework is grounded on minimal assumptions, enabling a direct foundation for G-Brownian motion and related stochastic calculus.