Risk Measure Induced By Generalized Relative Entropy And Its Robust Representation

Based on the relative entropy,entropic risk measure is a very important convex risk measure.Relative entropy theory are closely related to mathematical physics,mathematical finance,random control,risk measure,and have been widely and continuously studied.In this paper,inspired by the variational representation of relative entropy and entropic risk measure,based on generalized relative entropy theory,we study the variational representation of generalized relative entropy,generalized entropic risk measure and its robust representation,conditional generalized entropy risk measure and related backward stochastic differential equation.In the first chapter,it briefly introduces the research background,current situation and the main work of this paper.In the second chapter,generalized entropic risk measure is studied.Firstly,we briefly introduce the definition and properties of relative entropy and generalized entropy.Secondly,we establish a variational representation for the generalized relative entropy,conditional entropic risk measure is the solution of a class of backward stochastic differential equation(see Theorem 2.1).Next,by the variational representation result,the robust representation of generalized entropic risk measure is established,and we study related properties.Finally,we study the conditional generalized risk measure and prove the time consistency.In the third chapter,we consider the relationship between conditional generalized entropic risk measure and backward stochastic differential equation.Based on q-exponential transformation and q-logarithmic transformation,we obtain that conditional entropy risk measure is the solution of a class of backward stochastic differential equations(see Theorem 3.1).Furthermore,the robust control problem of conditional generalized entropic risk measure is considered by the dual method.In the fourth chapter,we summarizes and looks forward to the contents of this paper.