Abstract
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.
Highlights
In the paper, the notion of the regular super-martingale relative to the set of equivalent measures is introduced
The above method we use for the construction of evolution of risk assets and we describe completely the set of equivalent martingale measures for this evolution
We prove that every martingale measure is an integral on the set of extreme points of the convex set of martingale measures relative to some measure on it
Summary
The notion of the regular super-martingale relative to the set of equivalent measures is introduced. The sufficient conditions of the existence of the set of equivalent measures consistent with the filtration, satisfying the conditions: the mean value of the nonnegative random value relative to these set of measures equal one, are given. Using the above result we construct the example of the set of equivalent measures consistent with the filtration satisfying the condition: the mean value of the nonnegative random value relative to every measure of this set of measures equals one. For the integral from the nonnegative random value relative to all martingale measures which is dominated by one, the inequalities for this random value are obtained This fact gives us the possibility to find a new proof of the optional decomposition for the nonnegative super-martingale. We consider an application of the results obtained to find the new formula for the fair price of super-hedge in the case, as the risk asset evolves by the discrete geometric Brownian motion
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