A new method to transform systems of inequalities to find the interpolation of martingale measures

For one-step model of stochastic basis a random process $(Z_k, F_k)^1_{k=0}$ is considered where $F_0$ — is a trivial $\sigma$-field, and $F_1$ — is a $\sigma$-field generated by a countable number of atoms. It is assumed that $Z_1$ takes four different values $b_1<b_2<b_3<b_4$, where either $b_1<Z_0<b_2<b_3<b_4$ (this process admits an infinite number of martingale measures). This article takes up such martingale measures that satisfy the weakened noncoincidence barycenter condition (WNBC) — condition that makes it possible to interpolate with such martingale measure incomplete market to completel with respect to arbitrary interpolating special Haar filtering.
A new method of proof of the existence of the interpolation martingale measures is presented in the article. It is based on the replacement of complex inequalities of the WNBC that contains various undefined subsets of the set of natural numbers, into more simple inequalities containing specific components of martingale measures. In this article we obtain sufficient conditions on market parameters, which ensure the existence of a martingale measure.
The obtained results can be the basis for the algorithm and software complex. The program based on the method will allow to apply special Haar interpolations method to the calculations on the arbitrage-free financial markets, which will greatly facilitate the choice of optimal strategies of investors in the financial markets.