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Большой семинар кафедры теории вероятностей МГУ
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[Theorems, Method and Content of BSDE and Nonlinear Expectation Theory] Shige Peng Шаньдунский университет, КНР |
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Аннотация: In this talk, we present our research explorations of backward stochastic differential equations and nonlinear expectations. We first recall the basic definition of a space of linear, as well as nonlinear expectations, and then, through the representation theorem and examples of nonlinear i.i.d. (independent and identically distributed), and explain why this new framework can be applied to it’s own stochastic calculus and quantitatively analyze probabilistic and distributional dynamical uncertainty hidden behind data sequence. We have introduced two fundamentally important nonlinear Gaussian distribution and maxima distribution, then corresponding nonlinear law of large numbers (LLN) and nonlinear central limit theorem (CLT), which are crucial and fundamental breakthroughs. A typical application is a basic max-mean algorithm. We also present a basic continuous-time stochastic process, which is nonlinear Brownian motion (G-BM) and its stochastic calculus, including stochastic integral, stochastic differential equations, and the corresponding nonlinear martingale theory. This new theoretical framework has been deeply and strongly influenced by the axiomatic probability theory founded by Kolmogorov (1933). The key idea is to directly introduce the fundamental notion of nonlinear expectation Ê. The special case of the linear expectation corresponds a probability space (Ω, F, P). It is the nonlinearity that allows us to quantitatively measure the uncertainty of probabilities and probabilistic distributions inhabited in our real world data. Язык доклада: английский |
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