Weak Set-Valued Martingale Difference and Its Applications

Abstract

The martingale difference is considered widely in finance and economic because of its application in efficient market, in which the conditional expectation E[d t |F t−1 ] = 0 a.s., ∀t ≥ 2 for a sequence of asset returns {d t , t ≥ 1} and related historical information F t−1 . However, the concept of martingale difference in set-valued random variables (i.e. random sets) has not been studied. This paper proves some properties of a set-valued random variable sequence called a weak set-valued martingale difference. By studying its characteristic properties, we propose a method of testing the weak set-valued martingale difference hypothesis and perform some simulations with real data.