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(宋永生)非線性期望下伊藤過程分解的唯一性
2019-01-15 | 編輯:

  伊藤過程分解的唯一性是隨機分析最基本的結論之一,有著廣泛且重要的應用。特別地,它是概率方法研究偏微分方程的基礎。為了用概率方法研究PDE,首先要在概率空間中刻畫函數的導數。對于線性PDE,這等價于證明伊藤過程分解的唯一性,而對于非線性PDE,這等價于證明非線性期望下伊藤過程分解的唯一性。 

  非線性期望下伊藤過程分解的唯一性是非線性期望領域多年未解決的一個公開問題。這也是Peng and Song (2015)中的遺留問題。該問題的難點在于如何區分三類形式不同,但均可能具有絕對連續路徑的有限變差過程,因此傳統方法不能處理這類問題。   

  我們嚴格區分了非線性期望下的三類有限變差過程(1.有限變差鞅,2.,3. ),證明了非線性期望下伊藤過程分解的唯一性. 

  兩位審稿人對我們的結果和證明方法給出了積極評價: 

  Reviewer 1: 

  “In this paper the author has solved a deep and fundamental problem in the framework of stochastic analysis with G-Brownian motion B.” 

  “But as consequence, the above basic distinguishability problem becomes a challenging problem. 

  Some of the results and techniques introduced in this paper are also very original and useful for further research in this domain. 

  “I think this paper contains excellent and original results and thus is publishable in this journal.” 

  Reviewer 2: 

  “The obtained result is an important breakthrough, not only for the theory of stochastic analysis under G-expectation.” 

  “Let us emphasise that the proof of the main result, that of the uniqueness explained above, is rather tricky and technical, and full of new ideas. I like it very much.” 

  “For this reason there is no doubt for me that the submitted manuscript merits to be published. 

 

 

  發表論文    

  1. Song, Y. (2018): Properties of G-martingales with finite variation and the application to Sobolev spaces. Stochastic Processes and Their Applications. Available online at www.sciencedirect.com. 

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