Nonlinear Expectations and Stochastic Calculus under Uncertainty

TL;DR

This work develops a framework of sublinear expectations, introducing robust distributions, a new central limit theorem, and stochastic calculus tools to address uncertainty in probability models, with applications in finance and statistics.

Contribution

It introduces a novel sublinear expectation framework, including robust distributions, a new CLT, and stochastic calculus under uncertainty, advancing the mathematical tools for risk management.

Findings

Introduction of sublinear expectations for uncertainty modeling

Development of robust normal distributions and CLT under sublinear expectations

Formulation of stochastic calculus of Ito's type under this framework

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.