[Paper Review] $G$-martingale representation in the $G$-L'evy setting

Krzysztof Paczka|arXiv (Cornell University)|Apr 8, 2014

Stochastic processes and financial applications12 references5 citations

TL;DR

This paper establishes a martingale representation theorem for $G$-Lévy processes with finite activity and no drift, showing that any such martingale decomposes into three components: an Itô integral with respect to the continuous $G$-Brownian motion part, a compensated Itô-Lévy integral for the jump part, and a non-increasing continuous $G$-martingale. The key contribution is the rigorous decomposition under sublinear expectations, extending classical martingale representation to non-dominated probability measures with volatility and jump uncertainty.

ABSTRACT

In this paper we give the decomposition of a martingale under the sublinear expectation associated with a $G$-L'evy process X with finite activity and without drift. We prove that such a martingale consists of an Ito integral w.r.t. continuous part of a $G$-L'evy process, compensated Ito-L'evy integral w.r.t. jump measure associated with $X$ and a non-increasing continuous $G$-martingale starting at 0.

Motivation & Objective

• To extend martingale representation theory to $G$-Lévy processes, which generalize $G$-Brownian motion by incorporating jump uncertainty.
• To address model uncertainty in finance where volatility and jump measures are ambiguous and not dominated by a single probability measure.
• To establish a decomposition of $G$-martingales under sublinear expectations when drift is absent and volatility and jump uncertainties are independent.
• To prove that the decomposition includes a symmetric Itô integral, a compensated jump integral, and a non-increasing continuous $G$-martingale, with distinct properties under different probability measures.

Proposed method

• The framework uses a sublinear expectation space $(\Omega, \mathcal{H}, \mathbb{E})$ defined via viscosity solutions of non-linear integro-PDEs reflecting volatility, jump, and drift uncertainty.
• The analysis relies on the second-order non-degeneracy condition to ensure smoothness of viscosity solutions, enabling the use of Itô calculus in the $G$-Lévy setting.
• A compensation procedure is introduced for the Poisson random measure associated with jumps, proving that the compensated integral is a $G$-martingale under the sublinear expectation.
• A priori estimates are derived for the decomposition using Hölder’s inequality and essential suprema over a family of non-dominated probability measures $\mathfrak{P}$.
• The representation is first established for simple random variables via approximation and partitioning techniques over stopping times $\tau_\lambda$, then extended to a wider class of $L^p_G$-integrable random variables.
• The proof leverages the fact that the conditional expectation $M_t = \mathrm{ess\,sup}_{\mathbb{Q} \in \mathfrak{P}(t,\mathbb{P})} \mathbb{E}^\mathbb{Q}[\xi|\mathcal{F}_t]$ is well-defined and satisfies uniform bounds across $\mathbb{P} \in \mathfrak{P}$.

Experimental results

Research questions

• RQ1 How can a martingale under a $G$-Lévy process with finite activity and no drift be decomposed into fundamental components?
• RQ2 What is the role of the compensated Itô-Lévy integral in the martingale representation under sublinear expectations?
• RQ3 How do the different components of the decomposition behave under non-dominated probability measures?
• RQ4 Can the classical martingale representation theorem be extended to processes with both diffusion and jump uncertainty under $G$-expectation?
• RQ5 What conditions ensure the existence and uniqueness of such a decomposition in the $G$-Lévy setting?

Key findings

• The martingale decomposition consists of three orthogonal components: an Itô integral w.r.t. the continuous $G$-Brownian motion, a compensated Itô-Lévy integral for the jump measure, and a non-increasing continuous $G$-martingale starting at zero.
• The Itô integral component is a symmetric martingale, meaning it is a martingale under all probability measures in the family $\mathfrak{P}$, while the other components only satisfy supermartingale properties under some measures.
• The compensated integral w.r.t. the Poisson random measure is proven to be a $G$-martingale under the sublinear expectation, ensuring the validity of the decomposition.
• The decomposition holds for a wide class of $L^p_G$-integrable random variables, extending beyond simple processes via approximation and stopping time techniques.
• A uniform bound is established: $\mathbb{E}^\mathbb{P}[|M^*_T|^2] \leq C_p \|\xi\|_{L^p_G}^2$ for all $\mathbb{P} \in \mathfrak{P}$, with $C_p$ independent of $\mathbb{P}$, ensuring stability of the representation.
• The proof relies on the existence of a sequence $\{\mathbb{P}_j\}$ such that $M_{\tau_\lambda} = \sup_j \mathbb{E}^{\mathbb{P}_j}[\xi|\mathcal{F}_{\tau_\lambda}]$ $\mathbb{P}$-a.s., enabling the use of essential suprema and measure reweighting via $\hat{\mathbb{P}}^n$.