Research On Dynamic Monetary Risk Measures And G-Expectations

The theory of risk measures is a fruitful research area in the field of mathematical finance.The axiomatic based risk measures have been largely studied because most axioms possess desirable economic characteristics.The axioms of monotonicity and translation-invariance axioms have been largely accepted,but convexity limits the applicability of convex risk measures.Therefore,risk measures without convexity has attracted the attention of some scholars.Inspired by monetary risk measures(Jia-Xia-Zhao(2020))and star-shaped risk measures(Castagnoli-Cattelan-Maccheroni-Tebaldi(2022)),and taking into account dynamic information in risk assessment,establishing links between risk measures at different times,we study the properties and representation theorems of dynamic monetary risk measures and dynamic star-shaped risk measures.Based on the g-expectation theory,we study a class of time-consistent dynamic star-shaped risk measures.The structure of this paper is as follows.Chapter 1 briefly introduces the axiomatic development of risk measures,especially the development of non-convex risk measures and g-expectations,as well as the main research contents of this paper.In chapter 2,we study the representation theorems of dynamic monetary risk measures and dynamic star-shaped risk measures,and establish the link between dynamic convex risk measures and dynamic monetary risk measures by constructing acceptable sets.Similar to in static setting,a dynamic monetary(normalized star-shaped)risk measure can be expressed as the lower envelope of a family of dynamic(normalized)convex risk measures.We analyze the link between dynamic monetary risk measures and dynamic star-shaped risk measures.This chapter extends the static results of JiaXia-Zhao(2020)and Castagnoli-Cattelan-Maccheroni-Tebaldi(2022)to dynamics.Chapter 3 discusses a class of time-consistent normalized dynamic star-shaped risk measures induced by g-expectations.Under the generator gis star-shaped with respect to and a quadratic term in ,an equivalence relation between dynamic star-shaped risk measures and g-expectations is given.Under normality,we present the link between a class of time-consistent dynamic star-shaped risk measures and dynamic convex risk measures.We consider several examples of dynamic risk measures induced by BSDEs.Finally,we summarize the main contents of this paper.