| Abstract: |
| In this talk, we firstly address a fundamental challenge in mathematical finance and data-driven AI: how to measure and control risk under deep uncertainty, where a single probability distribution cannot be reliably determined from data. Then we introduce the theory of nonlinear expectations as a paradigm shift, presenting foundational limit laws-the nonlinear law of large numbers and the nonlinear central limit theorem-that characterize asymptotic behavior under model ambiguity. Building on this framework, a practical $\phi$-max-mean algorithm is developed for robust statistics and nonlinear distribution computation when the true data-generating distribution is unknown. We also show that phenomena governed by nonlinear expectations are pervasive, offering profound insights into quantitative risk management and the construction of reliable AI systems. By bridging deep mathematical theory with wide-ranging practical applications, this framework provides a unified and robust approach to decision-making under uncertainty. |
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