Generalized Stein's Lemma And Its Application In Risk Management

The application of Stein's lemma can be traced back to 1981,when it was used as a tool of Stein estimation.In the past decades,many scholars have found that the Stein's lemma also plays an important role in studying the risk measures of mixed moments.They apply the Stein's lemma to the risk theory,especially the tail condition expectation,the tail condition variance,the Wang premium principle,the multivariate tail condition expectation and the multivariate tail covariance matrix.This thesis mainly has done three aspects work.Firstly,we derive the expression of the mixed moment E[X12f(X)]for elliptical distributions.Secondly,we study the truncated Stein's lemma for generalized skew-elliptical distributions.Thirdly,we consider Stein type lemmas for location-scale mixture of generalized skew-elliptical distributions.Firstly,inspired by Stein's lemma,we use two different methods to derive E[X12 f(X)]for any measurable function f satisfying some regularity conditions.One is defined directly,the other is defined by Stein's lemma,and we obtain two different expressions.Then we prove that the two different expressions are equivalent.Moreover,using the result of elliptical distributions,we give the concrete expression of the mixed moment E[X12f(X)]for normal distribution,and by applying this result,we obtain the recurrence formula of the product moment of normally distributed random variable.We also give some examples of special product moments and compare them with other results.Later,we present other simplified expressions of the mixed moments E[X12 f(X)]for special distributions,such as multivariate Student-t,logistic and Laplace distributions.Secondly,inspired by truncated Stein's lemma of elliptical distributions and Stein's lemma of generalized skew-elliptical distributions,we consider Stein's lemma for truncated generalized skew-elliptical random vectors.We provide two forms for truncated generalized skew-elliptical random vectors.One is the general form and the other is a special form.Then,by taking skewed functions and density generators as some special functions,we obtain truncated Stein's lemmas corresponding to distributions,such as generalized skew-normal,skew normal,elliptical and normal distributions.In the latter case,under the model of generalized skew-elliptical distributions,using the previous results,we derive formulas of the conditional tail expectation for portfolio,the lower-orthant conditional tail expectation at probability level q,the upper-orthant conditional tail expectation at probability level q,the truncated version of Wang's premium,the multivariate tail conditional expectation and the multivariate tail covariance matrix.Finally,based on the work of established the Stein's lemma for generalized skew-elliptical random vectors,we derive Stein type lemmas for location-scale mixture of generalized skew-elliptical random vectors.We also consider some special cases such as the location-scale mixture of elliptical random vectors,the location-scale mixture of generalized skew-normal random vectors and the location-scale mixture of normal random vectors.As an application in risk theory,we give the results of the three-fund separation for generalized skew-elliptical and elliptical distributions.