Quasiconvex stochastic processes and a separation theorem

Dawid Kotrys,Kazimierz Nikodem

doi:10.1007/s00010-014-0279-1

Abstract

Quasiconvex stochastic processes are introduced. A characterization of pairs of stochastic processes that can be separated by a quasiconvex stochastic process and a stability theorem for quasiconvex processes are given.

Highlights

In [1], Baron et al proved that two real functions f and g defined on a real interval I can be separated by a convex function if and only if they fulfil the following inequality f λx + (1 − λ)y λg(x) + (1 − λ)g(y), for all x, y ∈ I and λ ∈ [0, 1]
He proved that two functions f, g : I → R can be separated by a quasiconvex function if and only if f λx + (1 − λ)y max g(x), g(y), for all x, y ∈ I and λ ∈ [0, 1]
In this paper we introduce the notion of quasiconvex stochastic processes and present some properties of them

Summary

Introduction

In [1], Baron et al proved that two real functions f and g defined on a real interval I can be separated by a convex function if and only if they fulfil the following inequality f λx + (1 − λ)y λg(x) + (1 − λ)g(y), for all x, y ∈ I and λ ∈ [0, 1]. In 1994 Smolarz [11] obtained an analogous result for quasiconvex functions. He proved that two functions f, g : I → R can be separated by a quasiconvex function if and only if f λx + (1 − λ)y max g(x), g(y) , for all x, y ∈ I and λ ∈ [0, 1] (see [3]). Our main result extends the Smolarz separation theorem to quasiconvex stochastic. As a consequence we obtain a Hyers-Ulam-type stability result for quasiconvex stochastic processes

Preliminaries

Main result

Hyers–Ulam stability