[Paper Review] Extended Conditional G-Expectations and Related Stopping Times

Mingshang Hu, Shigē Péng|arXiv (Cornell University)|Sep 16, 2013

Stochastic processes and financial applications15 references18 citations

TL;DR

This paper extends the theory of G-expectations by defining time-consistent conditional expectations for random variables beyond the classical L_G^1 space, using limits of monotone sequences. The key contribution is a rigorous extension to larger domains—L_G^{1*} and L_G^{1**}—preserving dynamical consistency and enabling the optional stopping theorem for G-expectations.

ABSTRACT

In this paper we extend the definition of time conditional G-expectations $\mathbb{\hat{E}}_{t}[\cdot]$ to a larger domain on which the dynamical consistency still holds. In fact we can consistently define, by taking the limit, the time conditional expectations for each random variable $X$ which is the downward limit (resp. upward limit) of a monotone sequence $\{X_{i}\}$ in $L_{G}^{1}(Ω)$. To accomplish this procedure, some careful analysis is needed. Moreover, we give a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional G-expectations.

Motivation & Objective

• Address the limitation of G-expectations in classical L_G^1 space, which excludes important random variables like stopped processes.
• Extend the domain of time conditional G-expectations to include downward and upward limits of monotone sequences in L_G^1(Ω).
• Ensure dynamical consistency (time consistency) is preserved in the extended framework.
• Define stopping times and prove the optional stopping theorem under the extended G-expectation framework.
• Establish a robust theoretical foundation for nonlinear expectations in the presence of Knightian uncertainty and path-dependent volatility.

Proposed method

• Define L_G^{1*}(Ω) as the space of random variables that are almost surely downward limits of sequences in L_G^1(Ω).
• Define L_G^{1**}(Ω) as the space of random variables that are almost surely upward limits of sequences in L_G^{1*}(Ω).
• Extend the conditional G-expectation Ẽ_t[⋅] via limit processes: Ẽ_t[X] = lim Ẽ_t[X_n] for monotone sequences X_n → X.
• Prove that the extended conditional expectation is well-defined and independent of the approximating sequence using convergence lemmas.
• Introduce a new class of stopping times adapted to the G-framework and prove the optional stopping theorem for extended G-expectations.
• Establish key properties such as monotonicity, continuity along monotone sequences, and the tower property for the extended conditional expectations.

Experimental results

Research questions

• RQ1 Can time conditional G-expectations be consistently extended beyond L_G^1(Ω) while preserving dynamical consistency?
• RQ2 How can one define stopping times and prove the optional stopping theorem in the extended G-expectation framework?
• RQ3 What are the necessary and sufficient conditions for the extended conditional G-expectation to be well-defined via limit processes?
• RQ4 Can the extended framework accommodate random variables arising from stopping continuous processes, such as exit times?
• RQ5 What properties (e.g., monotonicity, tower property) are preserved in the extended space L_G^{1**}(Ω)?

Key findings

• The extended conditional G-expectation Ẽ_t[X] is well-defined for any X ∈ L_G^{1*}(Ω) or X ∈ L_G^{1**}(Ω) as the limit of Ẽ_t[X_n] along monotone sequences.
• The extended conditional expectation preserves monotonicity: if X ≤ Y q.s., then Ẽ_t[X] ≤ Ẽ_t[Y] q.s.
• The extended conditional expectation is continuous along monotone sequences: if X_n ↓ X in L_G^{1*}(Ω), then Ẽ_t[X_n] ↓ Ẽ_t[X] q.s.
• The tower property holds: Ẽ_s[Ẽ_t[X]] = Ẽ_{s∧t}[X] for all X ∈ L_G^{1**}(Ω).
• The optional stopping theorem holds for stopping times in the extended G-framework, enabling the use of nonlinear martingales.
• The extended framework allows for the inclusion of random variables such as X_τ (the stopped process at exit time τ), which are not in L_G^1(Ω) but are in L_G^{1**}(Ω).