A New Central Limit Theorem under Sublinear Expectations (0803.2656v1)

Abstract: We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.

• The paper presents a new central limit theorem that extends classical probabilistic models using sublinear expectations to accommodate model uncertainty.
• It employs nonlinear PDE techniques and deep interior estimates to prove convergence of iid random variables to a G-normal distribution.
• The findings have significant applications in risk management, enhancing modeling strategies under uncertainty in finance and statistics.

A New Central Limit Theorem under Sublinear Expectations

The paper presents a novel exploration into sublinear expectation spaces, extending fundamental concepts such as the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) into this framework. These results have far-reaching implications, particularly for fields where model uncertainty plays a significant role, such as financial risk management and statistics.

Sublinear expectation is a powerful notion that generalizes the classical expectation by relaxing the additivity requirements, allowing for the consideration of uncertainty in the model parameters like mean and variance. This non-linear approach opens up new avenues for handling model uncertainty, a key challenge in various applied domains.

Framework and Key Concepts

Sublinear expectation spaces are constructed using a triple $(\Omega, H, \hat{E})$, where $\Omega$ denotes the sample space, $H$ the space of random variables, and $\hat{E}$ the sublinear expectation. This sublinear expectation satisfies four properties: monotonicity, constant preserving, sub-additivity, and positive homogeneity. These properties are analogous to those found in a traditional probability space but allow for greater flexibility in model specification.

The paper introduces G-distributions, encompassing G-normal distributions, that arise naturally within this sublinear framework. G-distributions are characterized by the G-heat equation, a form of parabolic partial differential equation (PDE), which is essential for describing their behavior under sublinear expectations. The connection between G-distributions and PDEs heavily relies on the viscosity solutions' theory, ensuring existence and uniqueness under given conditions.

Central Limit Theorem under Sublinear Expectation

A significant contribution of this research is the development of a Central Limit Theorem (CLT) in the context of sublinear expectations. It is shown that independent and identically distributed (iid) random variables, under sublinear expectations, converge to a G-normal distribution, even in the presence of mean and variance uncertainty. This generalization is critical in applications where such uncertainties cannot be ignored.

The theoretical justification for this CLT builds upon a new method that incorporates results from the theory of nonlinear PDEs, particularly using deep interior estimates. The assumptions underlying this result, such as the form of the sublinear expectation and distributional uncertainties, have potential for further refinement and expansion.

Implications and Future Directions

This CLT under sublinear expectations allows for a more robust modeling of financial instruments' risk and for developing more conservative and comprehensive statistical tools. Practically, this framework accommodates a broader class of models where traditional probabilistic assumptions of additivity do not hold, particularly important in financial mathematics for pricing under uncertainty and risk management.

Theoretically, this paper serves as a foundation for further exploration of stochastic calculus in sublinear spaces. Extensions of this work could include developing more refined simulations of sublinear stochastic processes or even extending G-distributions to higher-dimensional spaces and their applications.

This work's innovations lie not only in its mathematical framework but also in its implications for uncertainty quantifications in various applied fields. Future research might focus on leveraging these theoretical advancements to address practical challenges in economics, engineering, and beyond, where uncertainty is inherently non-linear and multi-faceted. The paper's findings stand as a significant step toward a comprehensive theory of probability under uncertainty, diverging from classical statistical models and paving the way for new research methodologies.