倒向随机微分方程下的算子表示及Jensen不等式的开题报告.docxVIP

倒向随机微分方程下的算子表示及Jensen不等式的开题报告

开题报告

题目：倒向随机微分方程下的算子表示及Jensen不等式

摘要：本文研究了倒向随机微分方程下的算子表示及Jensen不等式。在此基础上，运用随机积分理论，得到了该类方程的算子表示，并证明了Jensen不等式在该类方程中的适用性。具体而言，本文将首先介绍倒向随机微分方程及其解的定义，接着引入了算子表示的概念，并详细讨论了如何利用算子表示来表达解析函数。随后，本文将探讨倒向随机微分方程中的随机积分，定义Martingale型随机积分的等价关系并证明了其性质，利用Martingale型随机积分的理论，本文得到了倒向随机微分方程的算子表示。最后，本文将给出Jensen不等式的定义，并证明在倒向随机微分方程中其仍然适用。

关键词：倒向随机微分方程；算子表示；随机积分；Martingale型随机积分；Jensen不等式

Abstract:ThispaperstudiestheoperatorrepresentationandJensensinequalityunderthebackwardstochasticdifferentialequation.Basedonthestochasticintegraltheory,theoperatorrepresentationofthistypeofequationisobtained,andtheapplicabilityofJensensinequalityinthistypeofequationisproved.Specifically,thispaperwillfirstintroducethedefinitionofbackwardstochasticdifferentialequationsandtheirsolutions,andthenintroducetheconceptofoperatorrepresentation,anddiscussindetailhowtouseoperatorrepresentationtoexpressanalyticfunctions.Subsequently,thispaperwillexplorestochasticintegrationinbackwardstochasticdifferentialequations,definetheequivalencerelationofMartingale-typestochasticintegrationandproveitsproperties.BasedonthetheoryofMartingale-typestochasticintegration,thispaperobtainstheoperatorrepresentationofbackwardstochasticdifferentialequations.Finally,thispaperwillgivethedefinitionofJensensinequalityandprovethatitisstillapplicableinbackwardstochasticdifferentialequations.

Keywords:backwardstochasticdifferentialequations;operatorrepresentation;stochasticintegration;Martingale-typestochasticintegration;Jensensinequality