Nonlinear Expectations and Stochastic Calculus under Uncertainty (1002.4546v1)

Abstract: In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls.

• The paper introduces a novel framework using sublinear expectations to extend classical probability and model uncertainty.
• It develops G-calculus with tools like G-normal distributions and G-Brownian motion to capture variable variance in stochastic processes.
• The robust Central Limit Theorem and law of large numbers under uncertainty provide actionable insights for risk management in finance.

Nonlinear Expectations and Stochastic Calculus under Uncertainty

The paper "Nonlinear Expectations and Stochastic Calculus under Uncertainty" by Shige Peng presents a comprehensive exposition of probability models under uncertainty through the development and analysis of nonlinear expectations, with a particular focus on sublinear expectations. This work extends classical probability theory and introduces novel concepts such as $G$-normal distribution, $G$-Brownian motion, and related calculus, which offer robust methods for addressing risk and uncertainty in financial mathematics and other applied domains.

Sublinear Expectations and Their Representation

A central theme of this work is the concept of nonlinear expectations, particularly sublinear expectations, defined as monotone, constant preserving functionals on random variable spaces. Sublinear expectations are linked to measure uncertainty and are typically expressed as the supremum over a set of linear expectations. This framework is particularly useful in applications such as finance, where uncertain models of probability distributions lead to the need for robust risk assessment techniques.

Foundations of $G$-Calculus

Peng introduces $G$-calculus, which forms the theoretical basis for $G$-normal distributions and $G$-Brownian motion. Unlike classical normal distribution, the $G$-normal distribution encapsulates uncertainty in variance through sublinear functions, yielding distributions characterized by a set of possible variances. Similarly, $G$-Brownian motion generalizes classical Brownian motion by incorporating variance uncertainty directly into its definition, making it a powerful tool for modeling stochastic processes with ambiguous volatility.

Robust Central Limit Theorem and Law of Large Numbers

One of the pivotal results in the paper is the robust Central Limit Theorem (CLT) introduced under the framework of sublinear expectations. This CLT is derived without the stringent requirement of identical and independently distributed (i.i.d.) conditions, accommodating the uncertainty of distributions. The new law of large numbers further demonstrates how sums of certain random variables converge in distribution to maximal distributions, providing insights into the distributional behavior of aggregated uncertain processes.

Stochastic Process and Calculus

Peng constructs $G$-Brownian motions, showing their independent, identically distributed increments while satisfying non-standard calculus rules. Devised directly within the sublinear distribution framework, $G$-Brownian motions are equipped with quadratic variations and allow for the definition of a stochastic integral analogous to Itô’s integral. A significant property of the $G$-Brownian motion’s quadratic variation is its independent increment property relative to the past, highlighting its suitability for stochastic analysis under uncertainty.

Implications and Future Directions

This paper underscores the theoretical richness and flexibility afforded by nonlinear expectations and $G$-calculus in managing uncertainty. By providing tools such as the $G$-Expectations and the corresponding $G$-stochastic integrals, it opens multidisciplinary applications ranging from stochastic control, mathematical finance—particularly in pricing under model uncertainty—to robust statistics and risk management.

Moving forward, this foundational work invites further exploration into generalized $G$-stochastic processes, enhanced computational techniques for implementation in practical scenarios, and potential refinements in the mathematical treatment of capacity theory related to sublinear expectations. The notions introduced have set a precedent for addressing real-world problems involving ambiguity in probabilistic modeling, which classical methods might not effectively capture.

Conclusion

Shige Peng’s work on nonlinear expectations and stochastic calculus under uncertainty presents groundbreaking approaches to modeling and analyzing uncertainty. By extending classical probability into the domain of sublinear expectations and $G$-distributions, this paper offers robust and versatile tools for dealing with uncertainty in various practical and theoretical contexts, marking a significant step forward in probability theory and stochastic processes.