Description of Incomplete Financial Markets for Time Evolution of Risk Assets

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.


Introduction
In the paper, the notion of the regular super-martingale relative to the set of Proof. The proof of Lemma 1 see [12].

□
In the next Lemma, we present the formula for calculation of the conditional expectation relative to another measure from M. Lemma (7) are valid, where  P P n n P P E P P M P P (8) Proof. The proof of Lemma 2 is evident.   (9) belongs to the set M, where 2 M is a direct product of the set M by itself.  α ω is a consistent one with the filtration  n , if P is a measure on { } Proof. Suppose that ( ) 1 It is easy to see that ( ) Q P E s n n Q (14) The last equality follows from the equivalence of the measures 1 2 , Q Q and P. Altogether  , Ω  with the filtration  n on it, let there exist k equivalent measures 1 , , , 1 >  k P P k , and a nonnegative random value 0 1 ξ ≠ be such that Then, there exists the set of equivalent measures M consistent with the filtration  n , satisfying the condition where we took into account the equality 1 1 1 is valid for those random value ( ) ξ ω ψ ω α ω ω ω ω ξ ω ξ ω  ψ ω ψ ω ξ ω ξ ω α ω ω ω ω Then, for such ( ) 1 2 , α ω ω the equality (39) is true. Moreover, Let us prove the last statement of Lemma

Construction of the Regular Set of Measures
In the next two Lemmas, we investigate the closure of a convex set of equivalent measures presented in Lemma 5 by the formula (42). First, we consider the countable case.
Suppose that 1 Ω contains the countable set of elementary events and let 1  be a σ-algebra of all subsets of the set 1 Ω . Let 1 P be a measure on the σ-algebra 1  . We assume that ( )   , The distance between the measures ε Q and Let us prove the second part of Lemma 6. It is evident that the inequality is true. Due to arbitrariness of the small ε , Lemma 6 is proved. , , of the space 1 Ω we call exhaustive one if the following conditions are valid: 2) the ( ) -th decomposition is a sub-decomposition of the n-th one, that is, for every j, -th decomposition is a sub-decomposition of the n-th one, that is, for every , In the next Lemma we give the sufficient condition of the existence of exhaustive decomposition.
, Ω  be a measurable space with a complete separable metric space 1 Ω and Borel σ-algebra 1  on it. Then { } 1 1 , Ω  has an exhaustive decomposition.
the countable set of open balls as ε m runs all positive rational numbers, where ( ) let us consider the ball 8 . The point 0 ω belongs to this ball and for every ρ ω ω ρ ω ω ρ ω ω ρ ω ω is true. Therefore and the rational number , , ω ε = D B , and so on. We put that { } consists of two sets 1 D and 1 is constructed, then the set we construct from the various set of the kind , Ω  have an exhaustive decomposition and let ξ be an integrable random value relative to the measure P, satisfying the conditions (36). Then, the closure of the set of measure Q, given by the formula (42), relative to the pointwise convergence of measures contains the set of measures , ∈ x x R , of the set of real numbers when ( ) ξ ω ξ ω ω ω ω ω ξ ω ξ ω ξ ω ξ ω  P P , for every integrable finite valued random value ( ) 1 2 , , ω ω f with probability one, as → ∞ n , since it is a regular martingale. It is evident that for those , From the formula (90), we obtain From the formula (92), we obtain Lemma 8 is proved. □ The next Theorem 5 is a consequence of Lemma 5. if and only if as for We introduced above the following denotations: M is a nonempty one, since it contains those measures Q, for which the random value ( ) , is a local regular martingale.
, is a local regular martingale.
The sufficiency. Suppose that , is a local regular martingale. Let us prove that, if 1 2  ∈ Ω × Ω that have the full measure P P µ Proof. The where we introduced the denotation ( ) ( ) where , , 1. , , In the next Theorem 8, we assume that the random value ( ) 1 1 η ω is an integrable one.

Theorem 8. On the measurable space { }
, Ω  with the filtration n  on it, every measure Q of the regular set of measures M for the random value ( ) ( ) where the random value The equalities (120) mean that Then, this case is reduced to the previous one by the note that the sequence From this, it follows that Thus, T is a monotone class. But, 0 U T ⊂ . Hence, T contains the minimal monotone class generated by the algebra 0 U , that is,  , is the set of extreme points for the set M.
Let us introduce the denotations Note that the σ-algebra n  is generated by sets of the kind From this, for the measure Q the representation is true, where we introduced the denotations n P − and n P + which are the contractions of the measure n P onto the σ-algebras . ; It is evident that the expression (139) equals zero for every is a probability measure on the σ-algebra n  .
Taking into account the denotation (134) and the formula (143) we obtain that the measure ; . \ From the inequalities (158), we obtain the inequalities Consider the case b). From the inequality (159), we obtain the inequalities The inequalities (165) give the inequalities Ω be a complete separable metric space. Suppose that evolution of the risk asset is defined by the formula (114) and non risk asset evolve by the law 1, 0, If the nonnegative payment function N f is N  measurable integrable random value relative to every martingale measure and satisfying conditions Theorem 16 from [5], then the fair price of super-hedge is given by the formula Proof. The left part of the formula (175) for the super-hedge is true (see: [5]

Description of Local Regular Super-Martingales Relative to a Regular Set of Measures
In this section, we give the description of local regular super-martingales.
is a local regular super-martingale relative to the regular set of measures M on the measurable space { } , Ω  with the filtration n  on it. Proof. The proof is evident.
as m → ∞ . Passing to the limit in the last equality, as m → ∞ , we obtain .
Introduce into consideration a random value then the necessary and sufficient conditions for it to be a local regular one is belonging it to the set K.
Proof. The necessity is evident. Then, as m → ∞ . Passing to the limit in the last equality, as m → ∞ , we obtain ( ) 0 , .
Introduce into consideration a random value . Then,

Discrete Geometric Brownian Motion
In this section, we construct for the discrete evolution of risk assets the set of equivalent martingale measures and give a new formula for the fair price of super-hedge. Let ( ) , .
We assume that the evolution of non risk asset is given by the formula Then, the discount evolution of the risk asset we can rewrite in the form   Proof. Since 1 R is a separable metric space then due to Lemma 7 the Borel σ-algebra ( ) 1

B R
has the exhaustive decomposition. Therefore, the filtration , 0, n n N =  , has the exhaustive decomposition, due to Remark 1. Theorem 11 guarantee the formula for the fair price of super-hedge [5].

Conclusions
In the paper, we generalize the results of the paper [5]. Section 2 contains the de- Using Lemma 5, in Lemma 9, we construct an example of the set of equivalent measures consistent with the filtration.
Further we consider an evolution of risk asset with memory. In Theorem 8, we describe completely the set of martingale measures for the considered evolution and prove that every martingale measure of this family is an integral over some measure on the set of extreme points of the set of martingale measures. Theorem