In the paper, a class of discrete evolutions of risk
assets having the memory is considered. For such evolutions the description of
all martingale measures is presented. It is proved that every martingale
measure is an integral on the set of extreme points relative to some measure on
it. For such a set of evolutions of risk assets, the contraction of the set of martingale measures on the filtration is
described and the representation for it is found. The inequality for the
integrals from a nonnegative random value relative to the contraction of the
set of martingale measure on the filtration which is dominated by one is obtained. Using these inequalities a new proof of
the optional decomposition theorem for super-martingales is presented. The
description of all local regular super-martingales relative to the regular set
of measures is presented. The applications of the results obtained to
mathematical finance are presented. In the case, as evolution of a risk asset
is given by the discrete geometric Brownian motion, the financial market is
incomplete and a new formula for the fair price of super-hedge is founded.
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