TY - JOUR
T1 - The maximal variation of a bounded martingale
AU - Mertens, Jean Francois
AU - Zamir, Shmuel
PY - 1977/9
Y1 - 1977/9
N2 - 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}.
AB - 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}.
UR - http://www.scopus.com/inward/record.url?scp=51249182519&partnerID=8YFLogxK
U2 - 10.1007/BF02756487
DO - 10.1007/BF02756487
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AN - SCOPUS:51249182519
SN - 0021-2172
VL - 27
SP - 252
EP - 276
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 3-4
ER -