Semiparametric Bounds On Means And Variances Of Truncated Three-Piece Linear Functions

The subject of this paper is a three-piece linear convex function of a random variable, and the purpose of our research is to get the estimates of upper and lower bounds for several digital features of a truncated or untruncated three-piece linear convex function of this random variable under several moment conditions of it.This research belongs to the moment problem. Moment problem has a history of nearly two hundred years, and until now, it is still popular, for there are many new results, new approaches and new theories appearing in it all the time, and for it is widely applied in many branches of mathematics and many other related sciences.In many researches such as researches in economy and finance, the discussions of European options usually meet the estimates of digital features such as means for some functions of a random variable. In this paper, we first apply the duality approach to determine the semiparametric bounds for probabilities of a three-piece linear convex function of a random variable, with the best results of all five cases. Next, applying the same approach, we extend the known semiparametric bounds for means of three-piece linear convex function to the truncated case, with the best results of all seven cases instead of the former four cases. Last, as an important complement, we discuss the semiparametric bounds for variances of a truncated three-piece linear convex function with three estimates. The first two estimates are relatively easy and yet not obvious, but applying duality theory and metric approach, they are easily proved. In the third discussion, although there is one condition more, we still get a better estimate than the first two by constructing a proper controlling function.By giving profounder results, the research of this paper improves our former realization of semiparametric bounds for digital features such as means and variances of three-piece linear convex functions. Naturally, all the results in this paper can be applied to European options in the field of finance.