The maximal variation of a bounded martingale

Jean Francois Mertens*, Shmuel Zamir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

Let {Mathematical expression} be a martingale such that 0≦Xi≦1;i=0, ..., n. For 0≦p≦1 denote by ℳ p n the set of all such martingales satisfying also E(X0)=p. The variation of a martingale χ 0 n is denoted by V 0 n and defined by {Mathematical expression}. It is proved that {Mathematical expression}, where φ{symbol}(p) is the well known normal density evaluated at its p-quantile, i.e. {Mathematical expression}. A sequence of martingales χ 0 n, n=1,2, ... is constructed so as to satisfy {Mathematical expression}.

Original languageEnglish
Pages (from-to)252-276
Number of pages25
JournalIsrael Journal of Mathematics
Volume27
Issue number3-4
DOIs
StatePublished - Sep 1977

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