Martingales and Super-martingales Relative to a Convex Set of Equivalent Measures

In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.


Introduction
In the paper, a new method of investigation of martingales and super-martingales relative to a convex set of equivalent measures is developed. A new proof that the essential supremum over the set of regular martingales, generated by a certain nonnegative random value and a convex set of equivalent measures, is a super-martingale with respect to this set of measures, is given.
A notion of local regular super-martingale is introduced and the necessary and sufficient conditions are found under that the above defined super-martingales are local regular ones. The last fact allowed us to describe the local regular super-martingales. It is proved that the existence of a nontrivial martingale relative to a convex set of equivalent measures, generally speaking, not guarantee for a nonnegative super-martingale to be a local regular one.
An important notion of the complete convex set of equivalent measures is introduced. It is proved that any supermartingale relative to the complete convex set of equivalent measures on a measurable space with the finite set of elementary events is a local regular one. The notion of the complete convex set of equivalent measures is generalized onto an arbitrary space of elementary events. It is proved that the nonnegative and the majorized from below supermartingales are local regular ones.
The definition of the fair price of contingent claim is introduced. The sufficient conditions of the existence of the fair price of contingent claim are presented. The conditions that the introduced notion coincides with classical one are given. All these notions are used in the case as the convex set of equivalent measures is a set of equivalent martingale measures for the evolution of both risk and non risk assets. The formula for the fair price of Standard Contract with Option of European type in an incomplete market is found.
The notion of the complete convex set of equivalent measures permits us to give a new proof of the optional decomposition for a nonnegative super-martingale. This proof does not use the no-arbitrage arguments and the measurable choice [1], [2], [3], [4].
First, the optional decomposition for diffusion processes super-martingale was opened by by El Karoui N. and Quenez M. C. [5]. After that, Kramkov D. O. and Follmer H. [1], [2] proved the optional decomposition for the nonnegative bounded super-martingales. Folmer H. and Kabanov Yu. M. [3], [4] proved analogous result for an arbitrary super-martingale. Recently, Bouchard B. and Nutz M. [6] considered a class of discrete models and proved the necessary and sufficient conditions for the validity of the optional decomposition.
The optional decomposition for super-martingales plays the fundamental role for the risk assessment in incomplete markets [1], [2], [5], [7], [8], [9], [10]. Considered in the paper problem is a generalization of the corresponding one that appeared in mathematical finance about the optional decomposition for a super-martingale and which is related with the construction of the superhedge strategy in incomplete financial markets.
Our statement of the problem unlike the above-mentioned one and it is more general: a super-martingale relative to a convex set of equivalent measures is given and it is necessary to find the conditions for the super-martingale and the set of measures under that the optional decomposition exists.
The generality of our statement of the problem is that we do not require that the considered set of measures was generated by the random process that is a local martingale as it is done in the papers [1,4,5,6] and that is important for the proof of the optional decomposition in these papers.

Local regular super-martingales relative to a convex set of equivalent measures.
We assume that on a measurable space {Ω, F } a filtration F m ⊂ F m+1 ⊂ F , m = 0, ∞, and a family of convex set of equivalent measures M on F are given. Further, we assume that F 0 = { / 0, Ω} and the σ -algebra F = σ ( is a minimal σ -algebra generated by the algebra ∞ n=1 F n . A random process ψ = {ψ m } ∞ m=0 is said to be adapted one relative to the filtration {F m } ∞ m=0 if ψ m is a F m measurable random value, m = 0, ∞. Definition 1. An adapted random process f = { f m } ∞ m=0 is said to be a super-martingale relative to the filtration F m , m = 0, ∞, and the convex family of equivalent measures M if E P | f m | < ∞, m = 1, ∞, P ∈ M, and the inequalities Proof. Necessity. If { f m , F m } ∞ m=0 is a local regular super-martingale, then there exist a martingale {M m , F m } ∞ m=0 and a non-decreasing nonnegative random process {g m , F m } ∞ m=0 , g 0 = 0, such that From here we obtain the equalities where we introduced the denotationḡ 0 Sufficiency. Suppose that there exists an adapted nonnegative random processḡ 0 = {ḡ 0 m } ∞ m=0 ,ḡ 0 0 = 0, E Pḡ0 m < ∞, m = 1, ∞, such that the equalities (2) hold. Let us consider the random process It is evident that E P |M m | < ∞ and Theorem 1 is proved.
relative to a family of measures M for which there hold equalities E P f m = f 0 , m = 1, ∞, P ∈ M, is a martingale with respect to this family of measures and the filtration F m , m = 1, ∞.
3. Description of local regular super-martingales relative to a convex set of equivalent measures generated by the finite set of equivalent measures.
Below, we describe the local regular super-martingales relative to a convex set of equivalent measures M generated by the finite set of equivalent measures. For this we need some auxiliary statements.
Lemma 2. On a measurable space {Ω, F } with filtration F m on it, let G be a sub σ -algebra of the σ -algebra F and let f s , s ∈ S, be a finite family of nonnegative bounded random values. Then for every measure P from M Proof. We have the inequalities Therefore, The last implies In the next Lemma, we present the formula for calculation of the conditional expectation relative to another measure from M. Lemma 3. On a measurable space {Ω, F } with a filtration F n on it, let M be a convex set of equivalent measures and let ξ be a bounded random value. Then the following formulas are valid, where Proof. The proof of Lemma 3 is evident.
Let P 1 , . . . , P k be a family of equivalent measures on a measurable space {Ω, F } and let us introduce the denotation M for a convex set of equivalent measures Lemma 4. If ξ is an integrable random value relative to the set of equivalent measures P 1 , . . . , P k , then the formula is valid almost everywhere relative to the measure P 1 .
Proof. The definition of ess sup for non countable family of random variables see [13]. Using the formula or, On the other side [13], Therefore, Lemma 4 is proved.
Lemma 5. On a measurable space {Ω, F } with a filtration F n on it, let ξ be a nonnegative bounded random value. If dP i dP l , i, l = 1, k, are F 1 measurable and P 1 ( dP i dP l > 0) = 1, i, l = 1, k, then the inequalities are valid.
Proof. From Lemma 3 and Lemma 5 conditions relative to the density of one measure with respect to another, we have From the equality (21) we obtain the inequality Lemma 5 is proved.
In this section, we assume that the conditions of Lemma 5 relative to the density of one measure with respect to another are true. Lemma 6. On a measurable space {Ω, F } with a filtration F n on it, let ξ be a nonnegative random value which is integrable relative to the set of equivalent measures P 1 , . . . , P k . Then the inequalities are valid.
Proof. Using Lemma 5 inequalities for the nonnegative bounded ξ and the formula where Φ = max 1≤i≤k E P i {ξ |F n }, ϕ i = dP i dP 1 , i = 1, k, we prove Lemma 6 inequalities. Let us consider the case, as max 1≤i≤k E P i ξ < ∞. Let ξ s , s = 1, ∞, be a sequence of bounded random values converging to ξ monotonuosly. Then Due to the monotony convergence of ξ s to ξ , as s → ∞, we can pass to the limit under the conditional expectations on the left and right sides in the inequalities (25) that proves Lemma 6.
Lemma 7. On a measurable space {Ω, F } with filtration F n on it, for every nonnegative integrable random value ξ relative to a set of equivalent measures {P 1 , . . . , P k } the inequalities are valid.
Lemma 7 is a consequence of Lemma 6.
Lemma 8. On a measurable space {Ω, F } with a filtration F m on it, let ξ be a nonnegative integrable random value with respect to a set of equivalent measures {P 1 , . . . , P k } and such that then the random process {M m = ess sup P∈M E P {ξ |F m }, F m } ∞ m=0 is a martingale relative to a convex set of equivalent measures M.
Proof. Due to Lemma 7, a random process m=0 is a super-martingale, that is, Or, E P M m ≤ M 0 . From the other side, The above inequalities imply E P s M m = M 0 , m = 1, ∞, s = 1, k. The last equalities lead to the equalities E P M m = M 0 , m = 1, ∞, P ∈ M. The fact that M m is a super-martingale relative to the set of measures M and the above equalities prove Lemma 8, since the Lemma 1 conditions are valid.
In the next Theorem we denote Theorem 2. Let {Ω, F } be a measurable space with a filtration F m on it and let ξ be a nonnegative integrable random value with respect to a set of equivalent measures P 1 , . . . , P k . The necessary and sufficient conditions of the local regularity of the super-martingale is its uniform integrability relative to the set of measure P 1 , . . . , P k and the fulfillment of the equalities Proof. The necessity. Let { f m , F m } ∞ m=0 be a local regular super-martingale. Then From here we obtain E P i g n ≤ f 0 , i = 1, k. Due to the uniform integrability of f n and g n we obtain The last equality gives g ∞ = 0, or 4. Description of local regular super-martingales relative to an arbitrary convex set of equivalent measures.
Below, in the paper we assume that an arbitrary convex set of equivalent measures M on a measurable space {Ω, F } and a filtration F n on it satisfies the conditions: the density dP dQ is F 1 measurable one and P 0 ( dP dQ > 0) = 1 for all P, Q ∈ M, where the fixed measure P 0 ∈ M. Such a class of equivalent measures is sufficiently wide. It contains the class of equivalent martingale measures generated by a local martingale.
Introduce into consideration a set A 0 of all integrable nonnegative random values ξ relative to a convex set of equivalent measures M satisfying conditions It is evident that the set A 0 is not empty, since contains the random value ξ = 1. More interesting case is as A 0 contains more then one element.
Lemma 9. On a measurable space {Ω, F } and a filtration F n on it, let M be an arbitrary convex set of equivalent measures. If the nonnegative random value ξ is such that sup The set D = ∞ m=0 D m is also countable one and the equality ess sup is true. Really, since From the other side, The last gives ess sup The inequalities (38), (40) prove the needed statement. So, for all m we can choose the common set D. Let D = {P 1 , . . .P n , . . .}. Due to Lemma 7, for every Q ∈M k , we have and tending k to the infinity in the inequalities (41), we obtain The last inequalities implies that for every measure Q, belonging to the convex span, constructed on the set D, is a super-martingale relative to the convex set of equivalent measures, generated by the set D. Now, if a measure Q 0 does not belong to the convex span, constructed on the set D, then we can add it to the set D and repeat the proof made above. As a result, we proved that { f m = ess sup is also a super-martingale relative to the measure Q 0 . Zorn Lemma [14] complete the proof of Lemma 9.
Theorem 3. On a measurable space {Ω, F } and a filtration F n on it, let M be an arbitrary convex set of equivalent measures. For a random value ξ ∈ A 0 , the random process is a local regular martingale relative to the convex set of equivalent measures M.
Proof. Let P 1 , . . . , P n be a certain subset of measures from M. Denote M n a convex set of equivalent measures Let us consider an arbitrary measure P 0 ∈ M and let Then m=0 is a martingale relative to all measures from M. Due to Theorem 1, it is a local regular super-martingale with the random processḡ 0 m = 0, m = 0, ∞. Theorem 3 is proved.
then the random process is a local regular super-martingale relative to the convex set of equivalent measures M.
Proof. Due to Theorem 3, the random process is a martingale relative to the convex set of equivalent measures M. Therefore, It proves the needed statement.
m=0 is a local regular super-martingale relative to a convex set of equivalent measures M.
Denote F 0 the set of adapted processes For every ξ ∈ A 0 let us introduce the set of adapted processes and Corollary 2. Every random process from the set K, where is a local regular super-martingale relative to the convex set of equivalent measures M on a measurable space {Ω, F } with filtration F m on it.
Proof. The proof is evident.
Theorem 5. On a measurable space {Ω, F } and a filtration F n on it, let M be an arbitrary convex set of equivalent measures. Suppose that { f m , F m } ∞ m=0 is a nonnegative uniformly integrable super-martingale relative to a convex set of equivalent measures M, then the necessary and sufficient conditions for it to be a local regular one is belonging it to the set K.
i . Using the uniform integrability of f m , we can pass to the limit in the equality as m → ∞. Passing to the limit in the last equality, as m → ∞, we obtain Introduce into consideration a random value Let us putf 2 The same is valid forf 2 with ξ = 1. This implies that f belongs to the set K. Theorem 5 is proved. Theorem 6. On a measurable space {Ω, F } and a filtration F n on it, let M be an arbitrary convex set of equivalent measures. Suppose that the super-martingale { f m , F m } ∞ m=0 relative to the convex set of equivalent measures M satisfy conditions 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.
The inequalities f m +Cξ 0 ≥ 0, m = 1, ∞, give the inequalities From the inequalities (58) it follows that the super-martingale { f m , F m } ∞ m=0 is a uniformly integrable one relative to the convex set of equivalent measures M. The martingale {E P {ξ 0 |F m }, F m } ∞ m=0 relative to the convex set of equivalent measures M is also uniformly integrable one.
as m → ∞. Passing to the limit in the last equality, as m → ∞, we obtain Introduce into consideration a random value From here we obtain that ξ 1 ∈ A 0 and for the super is valid, where f 0 is a local regular one relative to the convex set of equivalent measures M. Corollary 3 is proved.

5.
Optional decomposition for super-martingales relative to the complete convex set of equivalent measures.
In this section we introduce the notion of complete set of equivalent measures and prove that non negative supermartingales are local regular ones with respect to this set of measures. For this purpose we are needed the next auxiliary statement.
Theorem 7. The necessary and sufficient conditions of the local regularity of the nonnegative super-martingale Proof. The necessity. Without loss of generality, we assume that f m ≥ a for a certain real number a > 0. Really, if it is not so, then we can come to the consideration of the super be a nonnegative local regular super-martingale. Then there exists a nonnegative adapted random process The sufficiency. Suppose that the conditions of Theorem 7 are valid. Then equality and inequalities give Let us consider the random process

Space of finite set of elementary events.
In this subsection we assume that a space of elementary events Ω is finite one, that is, N 0 = |Ω| < ∞, and we give a new proof of the optional decomposition for super-martingales relative to the complete convex set of equivalent measures. This proof does not use topological arguments as in [16].
Let F be a certain algebra of subsets of the set Ω and let F n ⊂ F n+1 ⊂ F be an increasing set of algebras, Denote M a convex set of equivalent measures on a measurable space {Ω, F }. Further, we assume that the set A 0 contains an element ξ 0 = 1. It is evident that every algebra F n is generated by sets Then for m n the representation is valid. Consider the difference d n (ω) = m n − m n−1 . Then where d n j ≤ 0, as j ∈ I − n , and d n j > 0 for j ∈ I + n . From the equalities (71), (72) we obtain Denote M n the contraction of the set of measures M on the algebra F n . Introduce into the set M n the metrics where B = {B n 1 , . . . , B n k } is a partition of Ω on k subsets, that is, The maximum in the formula (75) is all over the partitions of the set Ω, belonging to the σ -algebra F n .
for every i ∈ I − n and j ∈ I + n . Lemma 10. Let a convex family of equivalent measures M be a complete one and the set A 0 contains an element ξ 0 = 1. Then for every non negative F n -measurable random value ξ n = N n ∑ i=1 C n i χ A n i there exists a real number α n such that Proof. On the setM n , the functional ϕ(P) = N n ∑ i=1 C n i P(A n i ) is a continuous one, whereM n is the closure of the set M n in the metrics ρ n (P 1 , P 2 ). From this it follows that the equality is valid.
For those i ∈ I − n for which d n i < 0 and those j ∈ I + n for which d n j > 0 the inequality (79) is as follows From (80) we obtain the inequalities Since the inequalities (81) are valid for every , as d n i < 0, and since the set of such elements is finite, then if to denote then we have From the definition of α n we obtain the inequalities Now if d n i = 0 for some i ∈ I − n , then in this case f n i ≤ 1. All these inequalities give Multiplying on χ A n i the inequalities (85) and summing over all i ∈ I − n ∪ I + n we obtain the needed inequality. Lemma 10 is proved.
is a local regular one, where C n , n = 1, N, are constants.
It is evident that E P {ξ 0 n |F n−1 } = 1, P ∈ M, n = 1, N. Since sup P∈M E P ξ n ≤ 1, then Theorem 7 and the inequalities (88) prove Theorem 8. Proof. It is evident that every super-martingale { f m , F m } N m=0 is bounded. Therefore, there exists a constant C 0 > 0 such that 3C 0 2 > f m +C 0 > C 0 2 , ω ∈ Ω, m = 0, N. From this it follows that the super-martingale { f m +C 0 , F m } N m=0 is a nonnegative one and satisfies the conditions It implies that the conditions of Theorem 8 are satisfied. Theorem 9 is proved.
The last implies that P 0 ∈ M a 0 . Theorem 10 is proved.

Countable set of elementary events.
In this subsection, we generalize the results of the previous subsection onto the countable space of elementary events. Let F be a certain σ -algebra of subsets of the countable set of elementary events Ω and let F n ⊂ F n+1 ⊂ F be a certain increasing set of σ -algebras, where F 0 = { / 0, Ω}. Denote M a set of equivalent measures on the measurable space {Ω, F }. Further, we assume that the set A 0 contains an element ξ 0 = 1. Suppose that the σ -algebra F n is generated by the sets A n A n i = Ω, n = 1, ∞. Introduce into consideration the martingale m n = E P {ξ 0 |F n }, P ∈ M, n = 1, ∞. Then for m n the representation is valid. Consider the difference d n (ω) = m n − m n−1 . Then where d n j ≤ 0, as j ∈ I − n , and d n j > 0, j ∈ I + n . From the equalities (93), (94) we obtain Denote M n the contraction of the set of measures M on the σ -algebra F n . Introduce into the set M n the metrics where B = {B n 1 , . . . , B n k } is a partition of Ω on k subsets, that is, B n i ∈ F n , i = 1, k, B n i ∩ B n j = / 0, i = j, k i=1 B n i = Ω. The supremum in the formula (97 ) is all over the partitions of the set Ω, belonging to the σ -algebra F n . Definition 4. On a measurable space {Ω, F } with a filtration F n on it, a convex set of equivalent measure M we call complete one if for every 1 ≤ n < ∞ the closure of the set of measures M n in the metrics (97) contains the measures for every i ∈ I − n and j ∈ I + n .
Lemma 11. Let a family of measures M be complete and the set A 0 contains an element ξ 0 = 1. Then for every non negative bounded F n -measurable random value ξ n = ∞ ∑ i=1 C n i χ A n i there exists a real number α n such that Proof. On the setM n , the functional ϕ(P) = ∞ ∑ i=1 C n i P(A n i ) is a continuous one relative to the metrics ρ n (P 1 , P 2 ), wherē M n is the closure of the set M n in this metrics. From this it follows that the equality The last inequalities can be written in the form For those i ∈ I − n for which d n i < 0 and those j ∈ I + n for which d n j > 0 the inequality (102) is as follows From (103) we obtain the inequalities Two cases are possible: a) for all i ∈ I − n , f n i ≤ 1; b) there exists i ∈ I − n such that f n i > 1. First, let us consider the case a).
Since the inequalities (104) are valid for every , as d n i < 0, and f n i ≤ 1, i ∈ I − n , then if to denote we have 0 ≤ α n < ∞ and f n j ≤ 1 + α n d n j , d n j > 0, j ∈ I + n .
From the definition of α n we obtain the inequalities Now, if d n i = 0 for some i ∈ I − n , then in this case f n i ≤ 1. All these inequalities give Consider the case b). From the inequality (104), we obtain The last inequalities give Let us define α n = sup Then from (109) we obtain From the definition of α n , we have The inequalities (111), (112) give Multiplying on χ A n i the inequalities (108) and the inequalities (113) on χ A n j and summing over all i, j ∈ I − n ∪ I + n we obtain the needed inequality. The Lemma 11 is proved.
is a local regular one, where C m are constants.

Proof. From the conditions (114) it follows that sup
P∈M E P f m < ∞. Consider the random value ξ n = f n f n−1 . Due to Lemma 11 It is evident that E P {ξ 0 n |F n−1 } = 1, P ∈ M, n = 1, ∞. Since sup P∈M E P ξ n ≤ 1, then f n f n−1 ≤ ξ 0 n , n = 1, ∞.

An arbitrary space of elementary events.
In this subsection, we consider an arbitrary space of elementary events and prove the optional decomposition for non negative super-martingales. Let F be a certain σ -algebra of subsets of the set of elementary events Ω and let F n ⊂ F n+1 ⊂ F be an increasing set of the σ -algebras, where F 0 = { / 0, Ω}. Denote M a set of equivalent measures on a measurable space {Ω, F }. We assume that the σ -algebras F n , n = 1, ∞, and F are complete relative to any measure P ∈ M. Further, we suppose that the set A 0 contains an element ξ 0 = 1. Let m n = E P {ξ 0 |F n }, P ∈ M, n = 1, ∞.
For the random value d n (ω) there exists not more then a countable set of the real number d n s such that P(A n A n i ) > 0. Introduce for every n two subsets I − n = {ω ∈ Ω, d n (ω) ≤ 0}, I + n = {ω ∈ Ω, d n (ω) > 0} of the set {ω ∈ Ω, |d n (ω)| < ∞}. Denote M n the contraction of the set of measures M on the σ -algebra F n . Introduce into the set M n the metrics The supremum in the formula (117 ) is all over the partitions of the set Ω, belonging to the σ -algebra F n .
Lemma 12. Let a convex family of equivalent measures M be a complete one and the set A 0 contains an element ξ 0 = 1. Then for every non negative bounded F n -measurable random value ξ n there exists a real number α n such that ξ n sup P∈M E P ξ n ≤ 1 + α n (m n − m n−1 ), n = 1, ∞.
Proof. On the setM n , the functional ϕ(P) = Ω ξ n dP is a continuous one relative to the metrics ρ n (P 1 , P 2 ), whereM n is the closure of the set M n in this metrics. From this it follows that the equality sup P∈M n Ω ξ n dP = sup P∈M n Ω ξ n dP (120) is valid. Denote f n (ω) = ξ n (ω) sup P∈Mn E P ξ n (ω) . Then The last inequalities can be written in the form The inequality (122) for the measures (118) is as follows is a local regular one, where C m , m = 1, ∞, are constants.
Proof. From the inequalities (136) it follows that sup P∈M E P f m < ∞, m = 1, ∞. Consider the random value ξ n = f n f n−1 . Due to Lemma 12 It is evident that E P {ξ 0 n |F n−1 } = 1, P ∈ M, n = 1, ∞. Since sup P∈M E P ξ n ≤ 1, then Theorem 7 and the inequalities (138) prove Theorem 12.

is a nonnegative one and satisfies the conditions
From Theorem 11 it follows the validity of the local regularity for the super-martingale { f m + ε, F m } ∞ m=0 , therefore, for the super-martingale { f m , F m } ∞ m=0 the local regularity is also true.

Local regularity of majorized super-martingales.
In this section, we give the elementary proof that a majorized super-martingale relative to the complete set of equivalent measures is local regular one.
for certain constants 0 < C 1 ,C 2 < ∞, is a local regular one.

Application to Mathematical Finance.
Due to Corollary 3, we can give the following definition of the fair price of contingent claim f N relative to a convex set of equivalent measures M.
Definition 6. Let f N , N < ∞, be a F N -measurable integrable random value relative to a convex set of equivalent measures M such that for some 0 ≤ α 0 < ∞ and ξ 0 ∈ A 0 Denote the fair price of the contingent claim f N relative to a convex set of equivalent measures M, if there exists ζ 0 ∈ A 0 and a sequences α n ∈ [0, α 0 ], ξ α n ∈ A 0 , satisfying the conditions: α n → f 0 , ξ α n → ζ 0 by probability, as n → ∞, and such that Theorem 15. Let the set A 0 be uniformly integrable one relative to every measure P ∈ M. Suppose that for a nonnegative F N -measurable integrable contingent claim f N , N < ∞, relative to every measure P ∈ M there exist α 0 < ∞ and ξ 0 ∈ A 0 such that then the fair price f 0 of contingent claim f N exists. For f 0 the inequality Proof. If f 0 = α 0 , then Theorem 15 is proved. Suppose that f 0 < α 0 . Then there exists a sequence α n → f 0 , and ξ α n ∈ A 0 , n → ∞, such that Due to the uniform integrability A 0 we obtain Using again the uniform integrability of A 0 and going to the limit in (146) we obtain From the inequality is a local regular super-martingale, then From this it follows that f 0 = sup P∈M E P f N .
Let us prove that f 0 is a fair price for certain evolutions of risk and non risk assets. Suppose that the evolution of risk asset is given by the law S m = f 0 M P {ζ 0 |F m }, m = 0, N, and the evolution of non risk asset is given by the formula B m = 1, m = 0, N.
As proved above, for f 0 = inf α∈G α 0 α there exists ζ 0 ∈ A 0 such that the inequality is valid. Let us put It is evident thatf m−1 −f m ≥ 0, m = 0, N. Therefore, the super-martingale is a local regular one. It is evident that where For the martingale {M m , F m } N m=0 the representation is valid, where H i = 1, i = 1, N. Let us consider the trading strategy π = {H 0 m ,H m } N m=0 , wherē It is evident thatH 0 m ,H m are F m−1 measurable and the trading strategy π satisfy self-financed condition Moreover, the capital corresponding to the self-financed trading strategy π is given by the formula Herefrom, X π 0 = f 0 . Further, The last proves Theorem 15.
Theorem 16. Suppose that the set A 0 contains only 1 ≤ k < ∞ linear independent elements ξ 1 , . . . ξ k . If there exist ξ 0 ∈ T and α 0 ≥ 0 such that where then the fair price f 0 of the contingent claim f N ≥ 0 exists, where f N is F N measurable and integrable relative to every measure P ∈ M, N < ∞.
Proof. The proof is evident, as the set T is a uniformly integrable one relative to every measure from M. On a probability space {Ω, F , P}, let us consider an evolution of one risk asset given by the law {S m } N m=0 , where S m is a random value taking values in R 1 + . Suppose that F m is a filtration on {Ω, F , P} and S m is F m -measurable random value. We assume that the non risk asset evolve by the law B 0 m = 1, m = 1, N. Denote M e (S) the set of all martingale measures being equivalent to the measure P. We assume that the set M e (S) of such martingale measures is not empty and the effective market is non complete, see, for example, [15], [17], [18], [19]. So, we have that The next Theorem justify the Definition 6.
Theorem 17. Let a contingent claim f N be a F N -measurable integrable random value with respect to every measure from M e (S) and the conditions of the Theorem 16 are satisfied with ξ i = S i S 0 , i = 0, N. Then there exists self-financed trading strategy π the capital evolution {X π m } N m=0 of which is a martingale relative to every measure from M e (S) satisfying conditions X π 0 = f 0 , X π N ≥ f N , where f 0 is a fair price of contingent claim f N .
Proof. Due to Theorems 15, 16, for f 0 = inf α∈G α 0 α there exists ζ 0 ∈ A 0 such that the inequality is valid. Let us put It is evident thatf m−1 −f m ≥ 0, m = 0, N. Therefore, the super-martingale is a local regular one. It is evident that where Due to Theorem 20, for the martingale {M m } N m=0 the representation is valid. Let us consider the trading strategy π = {H 0 m ,H m } N m=0 , wherē It is evident thatH 0 m ,H m are F m−1 -measurable ones and the trading strategy π satisfy the self-financed condition Moreover, a capital corresponding to the self-financed trading strategy π is given by the formula Herefrom, X π 0 = f 0 . Further, Therefore X π N ≥ f N . Theorem 17 is proved.
In the next Theorem we assume that the evolutions of risk and non risk assets generate incomplete market [15], [17], [18], [19], [20], that is, the set of martingale measures contains more that one element.
The fair price of Standard European Put Option with the payment function f N = (K − S N ) + is given by the formula Proof. In Theorem 18 conditions, the set of equations E P ζ = 1, ζ ≥ 0, has the solutions ζ i = S i S 0 , i = 0, N. It is evident that α 0 = S 0 and ζ N = S N S 0 , since Let us prove the needed formula. Consider the inequality where Suppose that α satisfies the inequality If α satisfies additionally the equality then for all ω ∈ Ω (186) is valid. From (188) we obtain for α since D 2 N ≥ D 1 i . From here we obtain It is evident that α = f 0 satisfies the inequality (187). If D 2 N − K ≤ 0, then S N − K ≤ 0 and from (185) we can put α = 0. Then, the formula (186) is valid for all ω ∈ Ω. Let us prove the formula (183) for Standard European Put Option. If S N ≤ K it is evident that α 0 = K, and ζ 0 = 1, since Let us prove the needed formula. Consider the inequality Or, for S N ≤ K If α is a solution of the equality then for all ω ∈ Ω (194) is valid. From (195) we obtain for α Therefore, takes values in R d and H n is F n−1 -measurable random vector. Introduce into consideration a set of random values Lemma 13. The set of random values K 1 N is a closed subset in the set of finite valued random values L 0 (R 1 ) relative to the convergence by measure P ∈ M.
The proof of the Lemma 13 see, for example, [17]. Introduce into consideration a subset |H i n |. Let K N be a subset of the set K 1 Denote also a set where L ∞ + (Ω, F , P 0 } is a set of bounded nonnegative random values. LetC be the closure of C in L 1 (Ω, F , P 0 ) metrics.

Lemma 14.
If ζ ∈C and such that E P 0 ζ = 0, then for ζ the representation is valid for a certain finite valued predictable process H = {H n } N n=1 .
Proof. If ζ ∈ K N , then Lemma 14 is proved. Suppose that ζ ∈C, then there exists a sequence k n − f n , From here we obtain ||k n − ζ || P 0 ≤ 2||k n − f n − ζ || P 0 . Therefore, k n → ζ by measure P 0 . On the basis of Lemma 13, a set is a closed subset of L 0 (R 1 ) relative to the convergence by measure P 0 . From this fact, we obtain the proof of Lemma 14, since there exists the finite valued predictable process H ∈ H 0 such that for ζ the representation is valid.
. If for every Q ∈ M e (S), E Q ζ = 0, then there exists finite valued predictable process H such that for ζ the representation is valid.
Proof. If ζ ∈C, then (209) follows from Lemma 14. So, let ζ does not belong toC. As in Lemma 14,C is a closure of C in L 1 (Ω, F , P 0 ) metrics for the fixed measure P 0 . The setC is a closed convex set in L 1 (Ω, F , P 0 ). Consider the other convex closed set that consists from one element ζ . Due to Han -Banach Theorem, there exists a linear continuous functional l 1 , which belongs to L ∞ (Ω, F , P 0 ), and real numbers α > β such that and the inequalities l 1 (ζ ) > α, l 1 (ξ ) ≤ β , ξ ∈C, are valid. SinceC is a convex cone we can put β = 0. From the condition l 1 (ξ ) ≤ 0, ξ ∈C we have l 1 (ξ ) = 0, ξ ∈ K 1 N ∩ L 1 (Ω, F , P 0 ). From (210) and the inclusionsC ⊃ C ⊃ −L ∞ (Ω, F , P 0 ) we have q(ω) ≥ 0. Introduce a measure (211) Let us choose ξ = χ A (ω)(S j i − S j i−1 ), A ∈ F i−1 , where χ A (ω) is an indicator of a set A. We obtain So, Q * is a martingale measure that belongs to the set M a (S), which is a set of absolutely continuous martingale measures. Let us choose Q ∈ M e (S) and consider a measure Q 1 = (1 − γ)Q + γQ * , 0 < γ < 1. A measure Q 1 ∈ M e (S) and, moreover, E Q 1 ζ = γE Q * ζ > 0. We come to the contradiction with the conditions of Theorem 19, since for Q ∈ M e (S), E Q ζ = 0. So, ζ ∈C, and in accordance with Lemma 14, for ζ the declared representation in Theorem 19 is valid.
Theorem 20. For every martingale {M n , F n } ∞ n=0 relative to the set of measures M e (S), there exists a predictable random process H such that for M n , n = 0, ∞, the representation is valid.
Theorem 20 is proved.

Conclusions.
In the paper, we generalize Doob decomposition for super-martingales relative to one measure onto the case of super-martingales relative to a convex set of equivalent measures. For super-martingales relative to one measure for continuous time Doob's result was generalized in papers [21] [22]. Section 2 contains the definition of local regular super-martingales. Theorem 1 gives the necessary and sufficient conditions of the local regularity of super-martingale.
In spite of its simplicity, the Theorem 1 appeared very useful for the description of the local regular super-martingales. For this purpose we investigate the structure of super-martingales of special types relative to the convex set of equivalent measures, generated by a certain finite set of equivalent measures. The main result of the section 3 is Lemma 6, which allowed proving Lemma 8, giving the sufficient conditions of the existence of a martingale with respect to a convex set of equivalent measures generated by finite set of equivalent measures.
Theorem 2 describes all local regular non negative super-martingales of the special type (30) relative to the convex set of equivalent measures, generated by the finite set of equivalent measures.
In the Theorem 3, we give the sufficient conditions of the existence of the local regular martingale relative to an arbitrary set of equivalent measures and arbitrary filtration. After that, we present in Theorem 4 the important construction of the local regular super-martingales which we sum up in Corollary 2. Theorem 6 proves that every majorized super-martingale belongs to the described class (53) of the local regular super-martingales.
Theorem 7 gives a variant of the necessary and sufficient conditions of local regularity of non negative supermartingale relative to a convex set of equivalent measures. Definition 3 determines a class of the complete set of equivalent measures. Lemma 10 guarantees a bound (77) for all non negative random values allowing us to prove Theorem 8, stating that for every super-martingale the optional decomposition is valid. We extend the results obtained from the finite space of elementary events onto the case as a space of elementary events is a countable one. At last, the subsection 5.3 contains the generalization of the result obtained in subsection 5.2 onto the case of arbitrary space of elementary events. In section 6, we prove Theorems 13 and 14, stating that for every majorized super-martingale the optional decomposition is valid.
Corollary 3 contains the important construction of the local regular super-martingales playing the important role in the definition of the fair price of contingent claim relative to a convex set of equivalent measures. The Definition 6 is a fundamental one for the evaluation of risks in incomplete markets. Theorem 15 gives the sufficient conditions of the existence of the fair price of contingent claim relative to a convex set of equivalent measures. It also gives the sufficient conditions, when the defined fair price coincides with the classical value. In Theorem 16 the simple conditions of the existence of the fair price of contingent claim are given. In Theorem 17 we prove the existence of the self-financed trading strategy confirming the Definition 6 of the fair price as the parity between the long and short positions in contracts. As an application of the results obtained we prove Theorem 18, where the formulas for the Standard European Call and Put Options in an incomplete market we present. Section 8 contains auxiliary results needed for previous sections.