Theory, methods and meaning of nonlinear expectation theory

Abstract

This is a survey on the research developments of nonlinear expectation theory. We first recall the basic definition of a space of nonlinear expectation, and then, through the representation theorem and some examples of nonlinear i.i.d. (independent and identically distributed), to explain why this new framework can be applied to calculate and quantitatively analyze probabilistic and distributional uncertainty hidden behind a real world (possibly high-dimensional) data sequence. Then we introduce two fundamentally important nonlinear normal distribution and maxima distribution and the corresponding nonlinear law of large numbers and nonlinear central limit theorem, which are crucial and fundamental breakthroughs of this new research domain. A typical application is a basic algorithm named ``$\\varphi$-max-mean. We also present a basic continuous-time stochastic process---nonlinear Brownian motion and its stochastic calculus, including stochastic integral, stochastic differential equations, and the corresponding nonlinear martingale theory. This new theoretical framework has generalized the axiomatical probability theory founded by Kolmogorov (1933). The key difference is the notion of nonlinear expectation $\\hat{\\mathbb{E}}$, whose special linear case corresponds a probability space $(\\Omega,{\\mathcal{F}},P)$. It is the nonlinearity that allows us to quantitatively measure the uncertainty of probabilities and probabilistic distributions inhabited in our real world.