aa r X i v : . [ q -f i n . P R ] O c t Derivatives Pricing in Non-Arbitrage MarketN.S. Gonchar Bogolyubov Institute for Theoretical Physics of NAS of Ukraine.
Abstract
The general method is proposed for constructing a family of martingalemeasures for a wide class of evolution of risky assets. The sufficient condi-tions are formulated for the evolution of risky assets under which the familyof equivalent martingale measures to the original measure is a non-empty set.The set of martingale measures is constructed from a set of strictly nonneg-ative random variables, satisfying certain conditions. The inequalities areobtained for the non-negative random variables satisfying certain conditions.Using these inequalities, a new simple proof of optional decomposition the-orem for the nonnegative super-martingale is proposed. The family of spotmeasures is introduced and the representation is found for them. The condi-tions are found under which each martingale measure is an integral over theset of spot measures. On the basis of nonlinear processes such as ARCH andGARCH, the parametric family of random processes is introduced for whichthe interval of non-arbitrage prices are found. The formula is obtained forthe fair price of the contract with option of European type for the consideredparametric processes. The parameters of the introduced random processes areestimated and the estimate is found at which the fair price of contract withoption is the least.
Keywords:
Random process; Spot set of measures;Optional Doob decomposition; Super-martingale;Martingale; Assessment of derivatives.
The study of non-arbitrage markets was begun for the first time in Bachelier’s work[1]. Then, in the famous works of Black F. and Scholes M. [2] and Merton R. S. [3]the formula was found for the fair price of the standard call option of European type.The absence of arbitrage in the financial market has a very transparent economicsense, since it can be considered reasonably arranged. The concept of non arbitragein financial market is associated with the fact that one cannot earn money withoutrisking, that is, to make money you need to invest in risky or risk-free assets. Theexact mathematical substantiation of the concept of non arbitrage was first madein the papers [4], [5] for the finite probability space and in the general case in thepaper [6]. In the continuous time evolution of risky asset, the proof of absent ofarbitrage possibility see in [7]. The value of the established Theorems is that theymake it possible to value assets. They got a special name ”The First and TheSecond Fundamental Asset Pricing Theorems.” Generalizations of these Theoremsare contained in papers [8], [9], [10].If the martingale measure is not the only one for a given evolution of a riskyasset, then a rather difficult problem of describing all martingale measures arises inorder to evaluate, for example, derivatives.Assessment of risk in various systems was begun in papers [11], [12], [13], [14].Statistical studies of the time series of the logarithm of the price ratio of riskyassets contain heavy tails in distributions with strong elongation in the central re-gion. The temporal behavior of these quantities exhibits the property of clustering This work was partially supported by the Program of Fundamental Research of the Departmentof Physics and Astronomy of the National Academy of Sciences of Ukraine ”Mathematical modelsof non equilibrium processes in open systems” N 0120U100857.
Let { Ω N , F N , P N } be a direct product of the probability spaces { Ω i , F i , P i } , i =1 , N , Ω N = N Q i =1 Ω i , P N = N Q i =1 P i , F N = N Q i =1 F i , where the σ -algebra F N is a min-imal σ -algebra, generated by the sets N Q i =1 G i , G i ∈ F i . On the measurable space { Ω N , F N } , under the filtration F n , n = 1 , N , we understand the minimal σ -algebragenerated by the sets N Q i =1 G i , G i ∈ F i , where G i = Ω i for i > n. We also intro-duce the probability spaces { Ω n , F n , P n } , n = 1 , N , where Ω n = n Q i =1 Ω i , F n = n Q i =1 F i ,P n = n Q i =1 P i . There is a one-to-one correspondence between the sets of the σ -algebra F n , belonging to the introduced filtration, and the sets of the σ -algebra F n = n Q i =1 F i of the measurable space { Ω n , F n } , n = 1 , N . Therefore, we don’t introduce newdenotation for the σ -algebra F n of the measurable space { Ω n , F n } , since it alwayswill be clear the difference between the above introduced σ -algebra F n of filtrationon the measurable space { Ω N , F N } and the σ -algebra F n of the measurable space { Ω n , F n } , n = 1 , N . We assume that the evolution of risky asset { S n } Nn =0 , given on the probabil-ity space { Ω N , F N , P N } , is consistent with the filtration F n , that is, S n is a F n -measurable. Due to the above one-to-one correspondence between the sets of the σ -algebra F n , belonging to the introduced filtration, and the sets of the σ -algebra F n of the measurable space { Ω n , F n } , n = 1 , N , we give the evolution of risky assetsin the form { S n ( ω , . . . , ω n ) } Nn =0 , where S n ( ω , . . . , ω n ) is an F n -measurable randomvariable, given on the measurable space { Ω n , F n } . It is evident that such evolutionis consistent with the filtration F n on the measurable space { Ω N , F N , P N } . Further, we assume that P n (( ω , . . . , ω n ) ∈ Ω n , ∆ S n > > ,P n (( ω , . . . , ω n ) ∈ Ω n , ∆ S n < > , n = 1 , N , (1)where ∆ S n = S n ( ω , . . . , ω n ) − S n − ( ω , . . . , ω n − ) , n = 1 , N . − n = { ( ω , . . . , ω n ) ∈ Ω n , ∆ S n ≤ } , Ω + n = { ( ω , . . . , ω n ) ∈ Ω n , ∆ S n > } , (2)∆ S − n = − ∆ S n χ Ω − n ( ω , . . . , ω n ) , ∆ S + n = ∆ S n χ Ω + n ( ω , . . . , ω n ) , (3) V n ( ω , . . . , ω n − , ω n , ω n ) = ∆ S − n ( ω , . . . , ω n − , ω n ) + ∆ S + n ( ω , . . . , ω n − , ω n ) , ( ω , . . . , ω n − , ω n ) ∈ Ω − n , ( ω , . . . , ω n − , ω n ) ∈ Ω + n . (4)We use the following denotation Ω an , n = 1 , N , where a takes two values − and+ . Our assumption, in this paper, is that for Ω an , a = − , + , the representationsΩ − n = N n [ k =1 [ A ,k − n × V kn − ] , Ω + n = N n [ k =1 [ A ,k + n × V kn − ] , N n ≤ ∞ , (5)are true, whereΩ n − = N n [ k =1 V kn − , A ,k − n , A ,k + n ∈ F n , A ,k − n ∪ A ,k + n = Ω n ,A ,k − n ∩ A ,k + n = ∅ , V kn − ∩ V jn − = ∅ , k = j, V kn − ∈ F n − . (6)The number N n may be finite or infinite. Since Ω − n ∪ Ω + n = Ω n , Ω − n ∩ Ω + n = ∅ , and P n (Ω − n ) > , P n (Ω + n ) > , we have P n (Ω − n ) = N n X k =1 P n ( A ,k − n ) P n − ( V kn − ) ,P n (Ω + n ) = N n X k =1 P n ( A ,k + n ) P n − ( V kn − ) , P n ( A ,k − n ) + P n ( A ,k + n ) = 1 . (7)Further, in this paper, we assume that P n ( A ,k − n ) > , P n ( A ,k + n ) > , n =1 , N , k = 1 , N n . We also assume some technical suppositions: there exist subsets B ,k − n,i ∈ F n , i = 1 , I n , I n > , and B ,k + n,s ∈ F n , s = 1 , S n , S n > , satisfying theconditions B ,k − n,i ∩ B ,k − n,j = ∅ , i = j, B ,k + n,s ∩ B ,k + n,l = ∅ , s = l, k = 1 , N n ,P n ( B ,k − n,i ) > , i = 1 , I n , P n ( B ,k + n,s ) > , s = 1 , S n , k = 1 , N n ,A ,k − n = I n [ i =1 B ,k − n,i , A ,k + n = S n [ s =1 B ,k + n,s , k = 1 , N n . (8)4elow, we give the examples of evolutions { S n ( ω , . . . , ω n ) } Nn =1 for which therepresentations (5) are true.Suppose that the random values a i ( ω , . . . , ω i ) , η i ( ω i ) satisfy the inequalities0 < a i ( ω , . . . , ω i ) ≤ , η i ( ω i ) ≥ , P i ( η i ( ω i ) < > , P i ( η i ( ω i ) > > ,i = 1 , N . If S n ( ω , . . . , ω n ) is given by the formula S n ( ω , . . . , ω n ) = S n Y i =1 (1 + a i ( ω , . . . , ω i ) η i ( ω i )) , n = 1 , N , (9)then { ω i ∈ Ω i , η i ( ω i ) ≤ } = A , − i , { ω i ∈ Ω i , η i ( ω i ) > } = A , i ,V i − = Ω i − , Ω − i = A , − i × Ω i − , Ω + i = A , i × Ω i − , i = 1 , N . (10)In general case, let us consider the evolution of risky asset { S n ( ω , . . . , ω n ) } Nn =1 , givenby the formula S n ( ω , . . . , ω n ) = S n Y i =1 (1 + N i X k =1 η ki ( ω i ) χ V ki − ( ω , . . . , ω i − ) a ki ( ω , . . . , ω i )) , n = 1 , N , (11)where the random values a ki ( ω , . . . , ω i ) , η ki ( ω i ) satisfy the inequalities0 < a ki ( ω , . . . , ω i ) ≤ , η ki ( ω i ) ≥ , P i ( η ki ( ω i ) < > , P i ( η ki ( ω i ) > > ,i = 1 , N , k = 1 , N n , and N i S k =1 V ki − = Ω i − , V ki − ∩ V si − , k = s. Then, if to put { ω i ∈ Ω i , η ki ( ω i ) ≤ } = A ,k − i , { ω i ∈ Ω i , η ki ( ω i ) > } = A ,k + i , we obtainΩ − i = N i [ k =1 [ A ,k − i × V ki − ] , Ω + i = N i [ k =1 [ A ,k + i × V ki − ] , i = 1 , N . (12)∆ S n ( ω , . . . , ω n − , ω n ) ≤ , ( ω , . . . , ω n − , ω n ) ∈ Ω − n , , n = 1 , N , ∆ S n ( ω , . . . , ω n − , ω n ) > , ( ω , . . . , ω n − , ω n ) ∈ Ω + n , n = 1 , N . (13) In this section, we present the construction of the set of measures on the ba-sis of evolution of risky assets given by the formulas (9), (11) on the measur-able space { Ω N , F N } . For this purpose, we use the set of nonnegative randomvalues α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) , given on the probability space { Ω − n × Ω + n , F − n × F + n , P − n × P + n } , n = 1 , N , where F − n = F n ∩ Ω − n , F + n = F n ∩ Ω + n . The measure P − n is a contraction of the measure P n on the σ -algebra F − n and themeasure P + n is a contraction of the measure P n on the σ -algebra F + n . After that, we5rove that this set of measures, defined the above set of random values, is equiv-alent to the measure P N . At last, Theorem 1 gives the sufficient conditions underthat the constructed set of measures is a set of martingale measures for the con-sidered evolution of risky assets. Sometimes, we use the abbreviated denotations { ω , . . . , ω n } = { ω } n , { ω , . . . , ω n } = { ω } n . We assume that the set of random values α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) = α n ( { ω } n ; { ω } n ) , ( { ω } n ; { ω } n ) ∈ Ω − n × Ω + n , n = 1 , N , satisfies the following con-ditions: P − n × P + n (( { ω } n ; { ω } n ) ∈ Ω − n × Ω + n , α n ( { ω } n ; { ω } n ) >
0) = P n (Ω − n ) × P n (Ω + n ) , n = 1 , N ; (14) Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) dP n ( ω n ) < ∞ , ( { ω , . . . , ω n − } ; { ω , . . . , ω n − } ) ∈ Ω n − × Ω n − , ( ω , . . . , ω n − ) ∈ Ω n − , n = 1 , N ; (15) Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) dP n ( ω n ) dP n ( ω n ) = 1 , ( { ω , . . . , ω n − } ; { ω , . . . , ω n − } ) ∈ Ω n − × Ω n − , n = 1 , N . (16)In the next Lemma 1, we give the sufficient conditions under which the conditions(14) - (16) are valid. Lemma 1.
Suppose that for Ω an , a = − , + , n = 1 , N , the representations (5) aretrue. If the conditions inf ≤ k ≤ N n P n ( A ,k − n \ B ,k − n,i ) > , i = 1 , I n , I n > , n = 1 , N , inf ≤ k ≤ N n P n ( A ,k + n \ B ,k + n,s ) > , s = 1 , S n , S n > , n = 1 , N , inf ≤ k ≤ N n P n ( B ,k − n,i ) > , i = 1 , I n , I n > , n = 1 , N , inf ≤ k ≤ N n P n ( B ,k + n,s ) > , s = 1 , S n , S n > , n = 1 , N , Z Ω N ∆ S − n ( ω , . . . , ω n − , ω n ) dP N < ∞ , n = 1 , N , (17) are true, then the set of bounded random values α n ( { ω } n ; { ω } n ) , satisfying the con-ditions (14) - (16), is a nonempty set. roof. Let us put α i − n ( ω , . . . , ω n ) = N n X k =1 α − n,k,i ( ω n ) χ A ,k − n ( ω n ) χ V kn − ( ω , . . . , ω n − ) ,α s + n ( ω , . . . , ω n ) = N n X k =1 α + n,k,s ( ω n ) χ A ,k + n ( ω n ) χ V kn − ( ω , . . . , ω n − ) , where α − n,k,i ( ω n ) = (1 − δ ni ) χ B ,k − n,i ( ω n ) P n ( B ,k − n,i ) + δ ni χ A ,k − n \ B ,k − n,i ( ω n ) P n ( A ,k − n \ B ,k − n,i ) , < δ ni < , i = 1 , I n , k = 1 , N n , (18) α + n,k,s ( ω n ) = (1 − µ ns ) χ B ,k + n,s ( ω n ) P n ( B ,k + n,s ) + µ ns χ A ,k + n \ B ,k + n,s ( ω n ) P n ( A ,k + n \ B ,k + n,s ) , < µ ns < , s = 1 , S n , k = 1 , N n . (19)If to introduce the nonnegative set of real numbers γ i,s ≥ , i = 1 , I n , s = 1 , S n , I n ,S n X i,s =1 γ i,s = 1 , n = 1 , N , (20)then α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) = I n ,S n X i,s =1 γ i,s α i − n ( ω , . . . , ω n ) α s + n ( ω , . . . , ω n ) , n = 1 , N , (21)satisfies the condition (14) - (16).Really, due to the Lemma 1 conditions, the random values α n ( { ω } n ; { ω } n } ) ,n = 1 , N , are strictly positive by construction. Therefore, the conditions (14) aretrue.Due to the boundedness of α n ( { ω } n ; { ω } n } ) ≤ C, n = 1 , N , < C < ∞ , theinequalities Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) dP n ( ω n ) ≤ C Z Ω n ∆ S − n ( ω , . . . , ω n − , ω n ) dP n ( ω n ) < ∞ , n = 1 , N , (22)7re true for almost everywhere ( ω , . . . , ω n − ) ∈ Ω n − , n = 1 , N , relative to themeasure P n − , owing to the inequalities (17) and Foubini Theorem. This provesthe inequality (15). The equality (16) is also satisfied due to the construction of α n ( { ω } n ; { ω } n ) . Lemma 1 is proved.The values, which the random variables α n ( { ω } n ; { ω } n } ) , n = 1 , N , constructedin Lemma 1, take, are determined by the values at points ω n ∈ Ω − n and ω n ∈ Ω n for all ( ω , . . . , ω n − ) ∈ Ω n − . On the basis of the set of random values α n ( { ω } n ; { ω } n ) , n = 1 , N , constructedin Lemma 1, let us introduce into consideration the family of measure µ ( A ) on themeasurable space { Ω N , F N } by the recurrent relations µ ( ω ,...,ω N − ) N ( A ) = Z Ω N × Ω N χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × α N ( { ω , . . . , ω N − , ω N } ; { ω , . . . , ω N − , ω N } ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) (cid:21) dP N ( ω N ) dP N ( ω N ) , (23) µ ( ω ,...,ω n − ) n − ( A ) = Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) (cid:21) dP n ( ω n ) dP n ( ω n ) , n = 2 , N , (24) µ ( A ) = Z Ω × Ω χ Ω − ( ω ) χ Ω +1 ( ω ) α ( ω ; ω ) × (cid:20) ∆ S +1 ( ω ) V ( ω , ω ) µ ( ω )1 ( A ) + ∆ S − ( ω ) V ( ω , ω ) µ ( ω )1 ( A ) (cid:21) dP ( ω ) dP ( ω ) , (25)where we put µ ( ω ,...,ω N − ,ω N ) N ( A ) = χ A ( ω , . . . , ω N − , ω N ) , A ∈ F N . (26)8 emma 2. Suppose that the conditions of Lemma 1 are true. For the measure µ ( A ) , A ∈ F N , constructed by the recurrent relations (23) - (25), the representation µ ( A ) = Z Ω N N Y n =1 ψ n ( ω , . . . , ω n ) χ A ( ω , . . . , ω N ) N Y i =1 dP i ( ω i ) (27) is true and µ (Ω N ) = 1 , that is, the measure µ ( A ) is a probability measure beingequivalent to the measure P N , where we put ψ n ( ω , . . . , ω n ) = χ Ω − n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n )+ χ Ω + n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n ) , (28) ψ n ( ω , . . . , ω n − , ω n ) = Z Ω n χ Ω + n ( ω , . . . , ω n − , ω n ) α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − , (29) ψ n ( ω , . . . , ω n − , ω n ) = Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − . (30) Proof.
Due to Lemma 1 conditions, the set of the strictly positive bounded randomvalues α n ( { ω } n ; { ω } n ) , n = 1 , N , satisfying the conditions (14) - (16), is a non emptyset. We prove Lemma 2 by induction down. Let us denote µ ( ω ,...,ω N − ,ω N ) N ( A ) = χ A ( ω , . . . , ω N ) . (31)Then, Z Ω N ψ N ( ω , . . . , ω N − , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) dP N ( ω N ) = Z Ω N χ Ω − N ( ω , . . . , ω N − , ω N ) ψ N ( ω , . . . , ω N − , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) dP N ( ω N )+ Z Ω N χ Ω + N ( ω , . . . , ω N − , ω N ) ψ N ( ω , . . . , ω N − , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) dP N ( ω N ) =9 Ω N χ Ω − N ( ω , . . . , ω N − , ω N ) ψ N ( ω , . . . , ω N − , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) dP N ( ω N )+ Z Ω N χ Ω + N ( ω , . . . , ω N − , ω N ) ψ N ( ω , . . . , ω N − , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) dP N ( ω N ) . (32)Substituting ψ N ( ω , . . . , ω N − , ω N ) , ψ N ( ω , . . . , ω N − , ω N ) into (32), we obtain Z Ω N ψ N ( ω , . . . , ω N − , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) dP N ( ω N ) = Z Ω N × Ω N χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × α N ( { ω , . . . , ω N − , ω N } ; { ω , . . . , ω N − , ω N } ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) (cid:21) dP N ( ω N ) dP N ( ω N ) = µ ( ω ,...,ω N − ) N − ( A ) . (33)Suppose that we are proved that µ ( ω ,...,ω n − ,ω n ) n ( A ) = Z N Q i = n +1 Ω i N Y i = n +1 ψ i ( ω , . . . , ω i ) χ A ( ω , . . . , ω N ) N Y i = n +1 dP i ( ω i ) . (34)Let us calculate Z Ω n ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n ) = Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n )+ Z Ω n χ Ω + n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n ) =10 Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n )+ Z Ω n χ Ω + n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n ) . (35)Substituting ψ n ( ω , . . . , ω n − , ω n ) , ψ n ( ω , . . . , ω n − , ω n ) into (35), we obtain Z Ω n ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n ) = Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) (cid:21) dP n ( ω n ) dP n ( ω n ) . (36)From the recurrent relations (23) - (25), we have µ ( ω ,...,ω n − ) n − ( A ) = Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) (cid:21) dP n ( ω n ) dP n ( ω n ) , n = 1 , N . (37)From the last equality, we have µ ( ω ,...,ω n − ) n − = Z Ω n ψ n ( ω , . . . , ω n − , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) dP n ( ω n ) , n = 1 , N . (38)Substituting into (38) the induction supposition (34), we obtain µ ( ω ,...,ω n − ) n − ( A ) =11 N Q i = n Ω i N Y i = n ψ i ( ω , . . . , ω i ) χ A ( ω , . . . , ω N ) N Y i = n dP i ( ω i ) . (39)To prove that µ (Ω N ) = 1 , let us prove the equality Z Ω n ψ n ( ω , . . . , ω n ) dP n ( ω n ) = 1 , ( ω , . . . , ω n − ) ∈ Ω n − , n = 1 , N . (40)We have Z Ω n ψ n ( ω , . . . , ω n ) dP n ( ω n ) = Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) +∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) = Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) dP n ( ω n ) dP n ( ω n ) = 1 . (41)The last equality follows from the fact that the set of random values α n ( { ω } n ; { ω } n ) ,n = 1 , N , satisfies the condition (16). The equalities (40) proves that every measure(27), defined by the set of random values α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , n = 1 , N , satisfying the conditions (14), (16), is a probability measure being equivalent to themeasure P N . This proves Lemma 2
Note 1.
Due to the equality (40), the contraction of measure µ ( A ) , A ∈ F N , onthe σ -algebra F n of filtration we denote by µ n . If A belongs to the σ -algebra F n of filtration, then A = B × N Q i = n +1 Ω i , where B belongs to the σ -algebra F n of themeasurable space { Ω n , F n } , therefore, for this contraction we obtain the formula µ n ( A ) = Z Ω n n Y i =1 ψ i ( ω , . . . , ω i ) χ B ( ω , . . . , ω n ) n Y i =1 dP i ( ω i ) , B ∈ F n . (42) Further, we also use the probability spaces { Ω n , F n , µ n } , n = 1 , N , where under themeasure µ n ( B ) , B ∈ F n , we understand the measure, given by the formula µ n ( B ) = Z Ω n n Y i =1 ψ i ( ω , . . . , ω i ) χ B ( ω , . . . , ω n ) n Y i =1 dP i ( ω i ) , B ∈ F n . (43)12 ote 2. Assume that for α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) , constructed inLemma 1, the inequalities < c n ≤ α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) ≤ C n < ∞ , are true. Suppose that the conditions ∆ S − n ( ω , . . . , ω n − , ω n ) ≤ B n < ∞ , n = 1 , N , (44) are valid, where c n , C n , B n are constant, then the set of equivalent measures to themeasure P N , described in Lemma 2, is nonempty one.Proof. Due to Lemma 2 conditions, the equality (14) is true. Further, Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) dP n ( ω n ) ≤ B n , ( { ω , . . . , ω n − } ; { ω , . . . , ω n − } ) ∈ Ω n − × Ω n − , ( ω , . . . , ω n − ) ∈ Ω n − , Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) dP n ( ω n ) dP n ( ω n ) = 1 , ( { ω , . . . , ω n − } ; { ω , . . . , ω n − } ) ∈ Ω n − × Ω n − . (45)The last inequality and the equality (45) means that the conditions (14) - (16)are satisfied. Note 2 is proved.For the nonnegative random value α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , given on themeasurable space { Ω − n × Ω + n , F − n × F + n } , F − n = F n ∩ Ω − n , F + n = F n ∩ Ω + n , n = 1 , N , let us define the integral for the nonnegative random value f N ( ω , . . . , ω N ) relativeto the measure µ ( A ) using the recurrent relations µ f N n − ( ω , . . . , ω n − ) = Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ f N n ( ω , . . . , ω n − , ω n )+13 S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ f N n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) , n = 1 , N , (46) µ f N N − ( ω , . . . , ω N − ) = Z Ω N × Ω N χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × α N ( { ω , . . . , ω N − , ω N } ; { ω , . . . , ω N − , ω N } ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) f N ( ω , . . . , ω N − , ω N )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) f N ( ω , . . . , ω N − , ω N ) (cid:21) dP N ( ω N ) dP N ( ω N ) . (47)From the formula (27) of Lemma 2, it follows that E µ f N = Z Ω N N Y n =1 ψ n ( ω , . . . , ω n ) f N ( ω , . . . , ω N − , ω N ) N Y i =1 dP i ( ω i ) (48)for every nonnegative F N -measurable random value f N ( ω , . . . , ω N − , ω N ) . Theorem 1.
Suppose that the conditions of Lemma 1 are true. Then, the set ofnonnegative random values α n ( { ω } n ; { ω } n ) , n = 1 , N , satisfying the conditions E µ | ∆ S n ( ω , . . . , ω n − , ω n ) | = Z Ω N N Y i =1 ψ i ( ω , . . . , ω i ) | ∆ S n ( ω , . . . , ω n − , ω n ) | N Y i =1 dP i ( ω i ) < ∞ , n = 1 , N , (49) is a nonempty one and the convex linear span of the set of measures (27), definedby the random values α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , n = 1 , N , satisfying the con-ditions (49), is a set of martingale measures being equivalent to the measure P N . Proof.
Taking into account the equality (40), the conditions (49) can be written inthe form Z Ω N N Y i =1 ψ i ( ω , . . . , ω i ) | ∆ S n ( ω , . . . , ω n − , ω n ) | N Y i =1 dP i ( ω i ) = Z Ω n n Y i =1 ψ i ( ω , . . . , ω i ) | ∆ S n ( ω , . . . , ω n − , ω n ) | n Y i =1 dP i ( ω i ) =2 Z Ω n − n − Y i =1 ψ i ( ω , . . . , ω i ) Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) × dP n ( ω n ) dP n ( ω n ) n − Y i =1 dP i ( ω i ) , n = 1 , N . (50)Since the conditions of Lemma 1 are true, then the the set of bounded randomvalues α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , n = 1 , N , constructed in Lemma 1, satisfythe conditions (14) - (16).From the equality (50) for the set of bounded random values α n ( { ω } n ; { ω } n ) ,n = 1 , N , satisfying the conditions (14) - (16), we obtain the inequality Z Ω N N Y i =1 ψ i ( ω , . . . , ω i ) | ∆ S n ( ω , . . . , ω n − , ω n ) | N Y i =1 dP i ( ω i ) ≤ C Z Ω N ∆ S − n ( ω , . . . , ω n − , ω n ) dP N < ∞ , n = 1 , N , (51)for a certain constant 0 < C < ∞ . This proves that the set of nonnegative randomvalues α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , n = 1 , N , satisfying the conditions (49), is anon empty set.Let us prove that Z Ω n ψ n ( ω , . . . , ω n )∆ S n ( ω , . . . , ω n ) dP n ( ω n ) = 0 , ( ω , . . . , ω n − ) ∈ Ω n − , n = 1 , N . (52)Really, Z Ω n ψ n ( ω , . . . , ω n )∆ S n ( ω , . . . , ω n ) dP n ( ω n ) = Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) − ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) ∆ S − n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) ∆ S + n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) = 0 , (53)15ue to the condition (15).To complete the proof of Theorem 1, let A belongs to the filtration F n − , then A = B × N Q i = n Ω i , where B belongs to the σ -algebra F n − of the measurable space { Ω n − , F n − } . Taking into account the equality (41), (53), we have, due to Foubinitheorem, Z Ω N N Y i =1 ψ i ( ω , . . . , ω i ) χ A ( ω , . . . , ω N )∆ S n ( ω , . . . , ω n ) N Y i =1 dP i ( ω i ) = Z Ω n n Y i =1 ψ i ( ω , . . . , ω i ) χ B ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n ) n Y i =1 dP i ( ω i ) = Z Ω n − n − Y i =1 ψ i ( ω , . . . , ω i ) χ B ( ω , . . . , ω n − ) n − Y i =1 dP i ( ω i ) × Z Ω n ψ i ( ω , . . . , ω n )∆ S n ( ω , . . . , ω n ) dP n ( ω n ) = 0 . (54)The last means that E µ { S n ( ω , . . . , ω n ) |F n − } = S n − ( ω , . . . , ω n − ) . Since everymeasure, belonging to the convex linear span of the measures considered above, isa finite sum of such measures, then it is a martingale measure being equivalent tothe measure P N . Theorem 1 is proved.Our aim is to describe this convex span of martingale measures in particularcases.
In this section, we prove some inequalities, which will be very useful for to proveoptional decomposition for super-martingale relative to all martingale measures.First, we prove an integral inequality for a nonnegative random variable under thefulfillment of the inequality for this nonnegative random variable with respect tothe constructed family of measures µ ( A ) . Further, using this integral inequality forthe non-negative random variable, a pointwise system of inequalities is obtained forthis non-negative random variable for a particular case. After that, the pointwisesystem of inequalities is obtained for the non-negative random variable in the generalcase. Then, using the resulting pointwise system of inequalities, the inequality isestablished for this non-negative random variable whose right-hand side is such thatits conditional mathematical expectation is equal to one.
Definition 1.
Let { Ω , F } be a measurable space. The decomposition A n,k , n, k =1 , ∞ , of the space Ω we call exhaustive one, if the following conditions are valid:1) A n,k ∈ F , A n,k ∩ A n,s = ∅ , k = s, ∞ S k =1 A n,k = Ω , n = 1 , ∞ ;
2) the ( n + 1) -th decomposition is a sub-decomposition of the n -th one, that is, forevery j, A n +1 ,j ⊆ A n,k for a certain k = k ( j );
3) the minimal σ -algebra containing all A n,k , n, k = 1 , ∞ , coincides with F . emma 3. Let { Ω , F } be a measurable space with a complete separable metricspace Ω and Borel σ -algebra F on it. Then, { Ω , F } has an exhaustive decompo-sition. The proof of Lemma 3 see, for example, in [15], [16].For the proof of integral inequalities, we cannot require the fulfillment for therandom values α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , n = 1 , N , the condition (15) in theLemma 4. Lemma 4.
Suppose that Ω n is a complete separable metric space, F n is a correspond-ing Borel σ -algebra on Ω n , n = 1 , N , and the conditions of Lemma 1 are valid. If,on the probability space { Ω n − , F n − , µ n − } , for each B ∈ F n − , µ n − ( B ) > , thenonnegative random value f n ( ω , . . . , ω n − , ω n ) satisfies the inequality µ n − ( B ) Z B Z Ω n n Y i =1 ψ i ( ω , . . . , ω i ) f n ( ω , . . . , ω n ) n Y i =1 dP i ( ω i ) ≤ , B ∈ F n − , (55) then the inequality Z Ω n ψ n ( ω , . . . , ω n ) f n ( ω , . . . , ω n ) dP n ( ω n ) ≤ , { ω , . . . , ω n − } ∈ Ω n − , n = 1 , N , (56) is true almost everywhere relative to the measure P n − . Proof.
The metric space Ω n − is a complete separable metric space with the met-ric ρ ( x, y ) = n − P i =1 ρ i ( x i , y i ) , where x = ( x , . . . , x n − ) , y = ( y , . . . , y n − ) ∈ Ω n − , ( x i , y i ) ∈ Ω i , ρ i ( x i , y i ) is a metric in Ω i . This means that the metric spaceΩ n − has an exhaustive decomposition { B mk } ∞ m,k =1 . Suppose that ( ω , . . . , ω n − ) ∈ B m,k for a certain k, depending on m, and there exists an infinite numberof m for which µ n − ( B m,k ) > . On the probability space { Ω n − , F n − , µ n − } , for every integrable finite valued random value ϕ n − ( ω , . . . , ω n − ) the sequence E µ n − { ϕ n − ( ω , . . . , ω n − ) | ¯ F m } converges to ϕ n − ( ω , . . . , ω n − ) with probabilityone, as m → ∞ , since it is a regular martingale. Here, we denoted ¯ F m the σ -algebra, generated by the sets B m,k , k = 1 , ∞ . It is evident that for those B m,k , for which µ n − ( B m,k ) = 0 ,E µ n − { ϕ n − ( ω , . . . , ω n ) | ¯ F m } = R B m,k ϕ n − ( ω , . . . , ω n − ) dµ n − µ n − ( B m,k ) , ( ω , . . . , ω n ) ∈ B m,k . (57)Denote A m = A m ( ω , . . . , ω n − ) those sets B m,k for which ( ω , . . . , ω n ) ∈ B m,k fora certain k, depending on m, and µ n − ( A m ) >
0. Then, for every integrable finitevalued ϕ n − ( ω , . . . , ω n − )lim m →∞ R A m ϕ n − ( ω , . . . , ω n − ) dµ n − µ n − ( A m ) = ϕ n − ( ω , . . . , ω n − ) (58)17lmost everywhere relative to the measure µ n − . If to put ϕ n − ( ω , . . . , ω n − ) = Z Ω n ψ n ( ω , . . . , ω n ) f n ( ω , . . . , ω n ) dP n ( ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − , (59)then we obtain the proof of Lemma 4.In Theorem 2, we assume that for ∆ S n ( ω , . . . , ω n − , ω n ) , n = 1 , N , the repre-sentation ∆ S n ( ω , . . . , ω n − , ω n ) = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − , ω n ) η n ( ω n ) = d n ( ω , . . . , ω n − , ω n ) η n ( ω n ) , n = 1 , N , S > , (60)is true, where the random values d n ( ω , . . . , ω n − , ω n ) , a n ( ω , . . . , ω n − , ω n ) , η n ( ω n ) ,n = 1 , N , given on the probability space { Ω n , F n , P n } , satisfy the conditions0 < a n ( ω , . . . , ω n − , ω n ) ≤ , a n ( ω , . . . , ω n − , ω n ) η n ( ω n ) > ,d n ( ω , . . . , ω n − , ω n ) > , P n ( η n ( ω n ) > > , P n ( η n ( ω n ) < > . (61)From these conditions we obtain Ω − n = Ω − n × Ω n − , Ω + n = Ω n × Ω n − , whereΩ − n = { ω n ∈ Ω n , η n ( ω n ) ≤ } , Ω n = { ω n ∈ Ω n , η n ( ω n ) > } . From the suppositions above, it follows that P n (Ω − n ) > , P n (Ω n ) > . Themeasure P − n is a contraction of the measure P n on the σ -algebra F − n = Ω − n ∩ F n ,P n is a contraction of the measure P n on the σ -algebra F n = Ω n ∩ F n . Theorem 2.
Let Ω i be a complete separable metric space and let F i be a Borell σ -algebra on Ω i , i = 1 , N . Suppose that for ∆ S n ( ω , . . . , ω n − , ω n ) , n = 1 , N , therepresentation (60) is valid and Lemma 4 conditions are true. Then, for the non-negative random value f n ( ω , . . . , ω n − , ω n ) the inequalities χ Ω − n ( ω n ) χ Ω n ( ω n ) (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) ≤ , ( ω , . . . , ω n − ) ∈ Ω n − , ( ω n , ω n ) ∈ Ω − n × Ω n , n = 1 , N , (62) are true almost everywhere relative to the measure P n − × P − n × P n on the mea-surable space { Ω n − × Ω − n × Ω n , F n − × F − n × F n } . roof. Under Theorem 2 conditions, the set of martingale measures is a nonemptyone. Due to the equality (40), we obtain Z Ω N N Y i =1 ψ i ( ω , . . . , ω i ) f n ( ω , . . . , ω n ) N Y i =1 dP i ( ω i ) = Z Ω n n Y i =1 ψ i ( ω , . . . , ω i ) f n ( ω , . . . , ω n ) n Y i =1 dP i ( ω i ) . (63)Further, Z Ω n ψ n ( ω , . . . , ω n ) f n ( ω , . . . , ω n ) dP n ( ω n ) = Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) . (64) χ Ω − n ( ω , . . . , ω n ) = χ Ω n − ( ω , . . . , ω n − ) χ Ω − n ( ω n ) ,χ Ω + n ( ω , . . . , ω n ) = χ Ω n − ( ω , . . . , ω n − ) χ Ω n ( ω n ) . (65)Due to Lemma 4, the inequality Z Ω n Z Ω n χ Ω − n ( ω n ) χ Ω n ( ω n ) α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) ≤ , (66)is true almost everywhere relative to the measure P n − on the σ -algebra F n − . Letus put α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) = α n ( ω n ; ω n ) , (67)19here α n ( ω n ; ω n ) satisfy the condition Z Ω − n Z Ω n α n ( ω n ; ω n ) dP n ( ω n ) dP n ( ω n ) = 1 . (68)Since, on the probability space { Ω − n × Ω n , F − n × F n , P − n × P n } , there exists anexhaustive decomposition { A m,k } ∞ m,k =1 , let us put α n ( ω n ; ω n ) = (1 − ε ) χ A m,k ( ω n ; ω n ) µ n ( A m,k ) + ε χ Ω − n × Ω n \ A m,k ( ω n ; ω n ) µ n (Ω − n × Ω n \ A m,k ) , (69)where µ n ( A ) = [ P − n × P n ]( A ) , A ∈ F − n × F n , and we assume that µ n ( A m,k ) > ,µ n (Ω − n × Ω n \ A m,k ) > . Suppose that ( ω n ; ω n ) ∈ A m,k and µ n ( A m,k ) > m and k. Then, Z Ω n Z Ω n χ Ω − n ( ω n ) χ Ω n ( ω n ) " (1 − ε ) χ A m,k ( ω n ; ω n ) µ n ( A m,k ) + ε χ Ω − n × Ω n \ A m,k ( ω n ; ω n ) µ n (Ω − n × Ω n \ A m,k ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) ≤ . (70)Going to the limit as m, k → ∞ and then as ε → , we obtain the inequality χ Ω , − n ( ω n ) χ Ω , + n ( ω n ) (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) ≤ , ( ω , . . . , ω n − ) ∈ Ω n − . (71)which is valid almost everywhere relative to the measure µ n . Theorem 2 is proved.
Lemma 5.
Let Ω n be a complete separable metric space and let F n be a Borel σ -algebra on Ω n , n = 1 , N . If the conditions of Lemma 4 are true, then the inequality χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) ≤ , ( ω , . . . , ω n − ) ∈ Ω n − , (72) is valid almost everywhere relative to the measure P n − × [ P n × P n ] on the measurablespace { Ω n − × Ω n × Ω n , F n − × F n × F n } . roof. Due to the conditions for Ω an , a = − , + , the representationΩ an = N n [ k =1 [ A ,kan × V kn − ] (73)is true. Owing to Lemma 5 conditions, there exists an exhaustive decomposition D nmi , m, i = 1 , ∞ , such that ∞ S i =1 D nmi = Ω n , m = 1 , ∞ . Let us denote A ,kan ∩ D nmi = E nkami . It is evident that E nkami forms an exhaustive decomposition of sets A ,kan , n =1 , N , k = 1 , ∞ , a = − , + , correspondingly. Due to Lemma 4, the inequality Z Ω n ψ n ( ω , . . . , ω n ) f n ( ω , . . . , ω n ) dP n ( ω n ) ≤ , ( ω , . . . , ω n − ) ∈ Ω n − , (74)is true almost everywhere relative to the measure P n − . The equality Z Ω n ψ n ( ω , . . . , ω n ) f n ( ω , . . . , ω n ) dP n ( ω n ) = Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) (75)is valid. From the equality (75) and Lemma 4, the inequality Z Ω n Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) dP n ( ω n ) dP n ( ω n ) ≤ , (76)is true almost everywhere relative to the measure P n − on the σ -algebra F n − . Letus put α r,s − n ( ω , . . . , ω n ) = N n X k =1 α − n,k,r,s ( ω n ) χ A ,k − n ( ω n ) χ V kn − ( ω , . . . , ω n ) , m,i + n ( ω , . . . , ω n ) = N n X k =1 α + n,k,m,i ( ω n ) χ A ,k + n ( ω n ) χ V kn − ( ω , . . . , ω n − ) ,α r,s,m,in ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) = α r,s − n ( ω , . . . , ω n ) α m,i + n ( ω , . . . , ω n ) , (77)where α − n,k,r,s ( ω n ) = " (1 − δ ) χ E nk − rs ( ω n ) P n ( E nk − rs ) + δ χ A k − n \ E nk − rs ( ω n ) P n ( A k − n \ E nk − rs ) ,α + n,k,m,i ( ω n ) = " (1 − δ ) χ E nk + mi ( ω n ) P n ( E nk + mi ) + δ χ A k + n \ E nk + mi ( ω n ) P n ( A k + n \ E nk + mi ) , < δ < . (78)In the formulas (78), we assume that the inequalities P n ( E nk − rs ) > , P n ( A k − n \ E nk − rs ) > , P n ( E nk + mi ) > , P n ( A k + n \ E nk + mi ) > , (79)are true. Let us consider α r,s,m,in ( { ω , . . . , ω n − , ω n − } ; { ω , . . . , ω n − , ω n } ) = α r,s − n ( ω , . . . , ω n − , ω n ) α m,i + n ( ω , . . . , ω n − , ω n ) . (80)Suppose that ( ω , . . . , ω n − ) ∈ V kn − for a certain k. Then, α r,s,m,in ( { ω , . . . , ω n − , ω n − } ; { ω , . . . , ω n − , ω n } ) = " (1 − δ ) χ E nk − rs ( ω n ) P n ( E nk − rs ) + δ χ A k − n \ E nk − rs ( ω n ) P n ( A k − n \ E nk − rs ) × " (1 − δ ) χ E nk + mi ( ω n ) P n ( E nk + mi ) + δ χ A k + n \ E nk + mi ( ω n ) P n ( A k + n \ E nk + mi ) . (81)We assume that the point ( ω n , ω n ) ∈ E nk − rs × E nk + mi for the infinite number of r, s and m, i , where P n ( E nk − rs ) > , P n ( E nk + mi ) > . Substituting (81) into (76) and going to the limit as m, k → ∞ r, s → ∞ andthen as δ → , we obtain the needed inequality. Lemma 5 is proved. Theorem 3.
Suppose that the conditions of Theorem 2 are true. If for a certain ω n ∈ Ω − n and ω n ∈ Ω n the inequalities sup ( ω ,...,ω n − ) ∈ Ω n − S − n ( ω , . . . , ω n − , ω n ) < ∞ , sup ( ω ,...,ω n − ) ∈ Ω n − S + n ( ω , . . . , ω n − , ω n ) < ∞ , n = 1 , N , (82)22 re true, then the nonnegative random values f n ( ω , . . . , ω n − , ω n ) , n = 1 , N , satisfythe inequalities f n ( ω , . . . , ω n − , ω n ) ≤ (1 + γ n − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n − , ω n )) , n = 1 , N , (83) where γ n − ( ω , . . . , ω n − ) is a bounded F n − -measurable random value.Proof. From the inequality (71), it follows the inequality f n ( ω , . . . , ω n − , ω n ) ≤ − f n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) ∆ S + n ( ω , . . . , ω n − , ω n ) , ω n ∈ Ω − n , ω n ∈ Ω n . (84)Let us define γ n − ( ω , . . . , ω n − ) = inf { ω n ,η − n ( ω n ) > } − f n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) , (85)then, taking into account the inequality (84), we obtain the inequality f n ( ω , . . . , ω n − , ω n ) ≤ γ n − ( ω , . . . , ω n − )∆ S + n ( ω , . . . , ω n − , ω n ) . (86)From the definition of γ n − ( ω , . . . , ω n − ) , we obtain the inequality f n ( ω , . . . , ω n − , ω n ) ≤ − γ n − ( ω , . . . , ω n − )∆ S − n ( ω , . . . , ω n − , ω n ) . (87)The inequalities (86), (87) give the inequality f n ( ω , . . . , ω n − , ω n ) ≤ γ n − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n − , ω n ) . (88)Let us prove the boundedness of γ n − ( ω , . . . , ω n − ) . From the inequalities (86), (87)we obtain 1∆ S − n ( ω , . . . , ω n − , ω n ) ≥ γ n − ( ω , . . . , ω n − ) ≥ − S + n ( ω , . . . , ω n − , ω n ) . (89)Due to Theorem 3 conditions, we obtain the boundedness of γ n − ( ω , . . . , ω n − ) . The F n − measurability of the random value γ n − ( ω , . . . , ω n − ) follows from thefact that Ω n is separable metric space and infimum is reached on the countable set,which is dense in Ω n . Theorem 3 is proved.
Theorem 4.
Let the conditions of Lemma 5 be valid. If there exist ω n ∈ A k − n , ω n ∈ A k + n , and the real numbers a k , b k , k = 1 , N n , such that sup ( ω ,...,ω n − ) ∈ V kn − S − n ( ω , . . . , ω n − , ω n ) = a nk < ∞ , ( ω ,...,ω n − ) ∈ V kn − S + n ( ω , . . . , ω n − , ω n ) = b nk < ∞ , k = 1 , N n , n = 1 , N , max ≤ n ≤ N sup ≤ k ≤ N n max { a nk , b nk } < ∞ , (90) then there exists a bounded F n − -measurable random value γ n − ( ω , . . . , ω n − ) suchthat the inequalities f n ( ω , . . . , ω n − , ω n )) ≤ (1 + γ n − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n − , ω n )) , n = 1 , N , (91) are true.Proof. For ω n ∈ A k − n , ω n ∈ A k + n and ( ω , . . . , ω n − ) ∈ V kn − , we have that( ω , . . . , ω n − , ω n ) ∈ Ω − n , ( ω , . . . , ω n − , ω n ) ∈ Ω + n . Then, from the inequality (72),we obtain the inequality (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f n ( ω , . . . , ω n − , ω n ) (cid:21) ≤ . (92)From the inequality (92), it follows the inequality f n ( ω , . . . , ω n − , ω n ) ≤ − f n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) ∆ S + n ( ω , . . . , ω n − , ω n ) . (93)Let us define γ kn − ( ω , . . . , ω n − ) =inf { ω n ∈ A ,k − n } − f n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) , ( ω , . . . , ω n − ) ∈ V kn − , (94)then, taking into account the inequality (93), we have the inequality f n ( ω , . . . , ω n − , ω n ) ≤ γ kn − ( ω , . . . , ω n − )∆ S + n ( ω , . . . , ω n − , ω n ) . (95)From the definition of γ kn − ( ω , . . . , ω n − ) , we obtain the inequality f n ( ω , . . . , ω n − , ω n ) ≤ − γ kn − ( ω , . . . , ω n − )∆ S − n ( ω , . . . , ω n − , ω n ) . (96)The inequalities (95), (96) give the inequality f n ( ω , . . . , ω n − , ω n ) ≤ γ kn − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n − , ω n ) . (97)Let us prove the boundedness of γ kn − ( ω , . . . , ω n − ) . From the inequalities (95), (96),we obtain the inequalities 24 nk = sup ( ω ,...,ω n − ) ∈ V kn − S − n ( ω , . . . , ω n − , ω n ) ≥ γ kn − ( ω , . . . , ω n − ) ≥ − sup ( ω ,...,ω n − ) ∈ V kn − S + n ( ω , . . . , ω n − , ω n ) = − b nk . (98)From this, it follows the boundedness of γ kn − ( ω , . . . , ω n − ) . The F n − measurabilityof the random value γ kn − ( ω , . . . , ω n − ) follows from the fact that Ω n is separablemetric space and infimum is reached on the countable set, which is dense in Ω n . Tocomplete the proof of Theorem 4, let us put γ n − ( ω , . . . , ω n − ) = N n X k =1 χ V kn − (( ω , . . . , ω n − ) γ kn − ( ω , . . . , ω n − ) , (99)then for such γ n − ( ω , . . . , ω n − ) the inequality (91) are satisfied. Theorem 4 isproved. In this section, we give simple proof of optional decomposition for the nonnegativesuper-martingale relative to the set of equivalent martingale measures. Such a prooffirst appeared in the paper [16]. First, the optional decomposition for diffusionprocesses super-martingale was opened by El Karoui N. and Quenez M. C. [21]. Afterthat, Kramkov D. O. and Follmer H. [22], [23] proved the optional decomposition forthe nonnegative bounded super-martingales. Folmer H. and Kabanov Yu. M. [24],[25] proved analogous result for an arbitrary super-martingale. Recently, BouchardB. and Nutz M. [26] considered a class of discrete models and proved the necessaryand sufficient conditions for the validity of the optional decomposition.
Theorem 5.
Let Ω i be a complete separable metric space and let F i be a Borell σ -algebra on Ω i , i = 1 , N . Suppose that the evolution { S n ( ω , . . . , ω n ) } Nn =1 of riskyassets satisfies the conditions of Theorems 1, 2, 3, 4, then for every nonnegativesuper-martingale { f n ( ω , . . . , ω n ) } Nn =0 relative to the set of martingale measure M, described in Theorem 1, the optional decomposition is true.Proof. Without loss of generality, we assume that f n ( ω , . . . , ω n ) ≥ a, where a isa real positive number. If it is not so, then we can come to the super-martingale f n ( ω , . . . , ω n ) + a. Let us consider the set of random values f n ( ω , . . . , ω n ) = f n ( ω , . . . , ω n ) f n − ( ω , . . . , ω n − ) , n = 1 , N . (100)Every random value f n ( ω , . . . , ω n ) satisfies the conditions of Lemma 4. Due toTheorems 3, 4, the inequalities f n ( ω , . . . , ω n ) f n − ( ω , . . . , ω n − ) ≤ γ n − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n ) , n = 1 , N , (101)25re true, where γ n − ( ω , . . . , ω n − ) is a bounded F n − -measurable random value.Since E Q | ∆ S n ( ω , . . . , ω n ) | < ∞ , Q ∈ M, we have E Q { γ n − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n ) |F n − } = 0 , Q ∈ M, n = 1 , N . (102)Let us denote ξ n ( ω , . . . , ω n ) = 1 + γ n − ( ω , . . . , ω n − )∆ S n ( ω , . . . , ω n ) , n = 1 , N . (103)Then, from the inequalities (101), we obtain the inequalities f n ( ω , . . . , ω n ) ≤ f n − ( ω , . . . , ω n − ) + f n − ( ω , . . . , ω n − )[ ξ n ( ω , . . . , ω n ) − , n = 1 , N . (104)Introduce the denotations g n ( ω , . . . , ω n ) = − f n ( ω , . . . , ω n ) + f n − ( ω , . . . , ω n − ) ξ n ( ω , . . . , ω n ) , n = 1 , N . (105)Then, g n ( ω , . . . , ω n ) ≥ , n = 1 , N , and E Q g n ( ω , . . . , ω n ) ≤ E Q f n ( ω , . . . , ω n ) + E Q f n ( ω , . . . , ω n − ) . (106)The equalities (105) give the equalities f n ( ω , . . . , ω n ) = f + n X i =1 f i − ( ω , . . . , ω n − )[ ξ i ( ω , . . . , ω i ) − − n X i =1 g i ( ω , . . . , ω i ) , n = 1 , N . (107)Let us put M n ( ω , . . . , ω n ) = f + n X i =1 f i − ( ω , . . . , ω i − )[ ξ i ( ω , . . . , ω i ) − , n = 1 , N , (108)then E Q { M n ( ω , . . . , ω n ) |F n − } = M n − ( ω , . . . , ω n − ) . Theorem 5 is proved.
In this section, we introduce the family of spot measures. After that, we obtainthe representations for the family of spot measures and define integral over theseset of measures. The sufficient conditions are found, under which the integral overthese set of measures is a set of martingale measures being equivalent to the initialmeasure. The introduced family of spot measures is a family of extreme points forthese set of equivalent measures.We give an evident construction of the set of martingale measures for riskyassets evolution, given by the formula (9). First of all, to do that we consider a26imple case as the measures P n is concentrated at two points ω n , ω n ∈ Ω n , where ω n ∈ A k − n , ω n ∈ A k + n for a certain k, depending on n, for the representation Ω − n and Ω + n , given by the formula (5). Let us put P n ( ω n ) = p kn , P n ( ω n ) = 1 − p kn , where0 < p kn < . Then, to satisfy the conditions (14) - (16), we need to put α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) = 1 p kn (1 − p kn ) , n = 1 , N , (109)and to require that∆ S − n ( ω , . . . , ω n − , ω n ) < ∞ , ( ω , . . . , ω n − , ω n ) ∈ Ω − n , ∆ S + n ( ω , . . . , ω n − , ω n ) < ∞ , ( ω , . . . , ω n − , ω n ) ∈ Ω + n . (110)Let us denote µ { ω n ,ω n } ,..., { ω N ,ω N } ( A ) the measure, generated by the recurrent relations(23) - (25), for the measures P n , n = 1 , N , concentrated at two points. For the point { ω n , ω n } , . . . , { ω N , ω N } ∈ Ω N × Ω N , the recurrent relations (23) - (25) is convertedrelative to the set of measures µ ( ω ,...,ω n − ) { ω n ,ω n } ,..., { ω N ,ω N } ( A ) into the recurrent relations µ ( ω ,...,ω N − ) { ω N ,ω N } ( A ) = χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) µ ( ω ,...,ω N − ,ω N ) N ( A ) (cid:21) , A ∈ F N , (111) µ ( ω ,...,ω n − ) { ω n ,ω n } ,..., { ω N ,ω N } ( A ) = χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A ) (cid:21) , n = 2 , N , A ∈ F N , (112) µ { ω n ,ω n } ,..., { ω N ,ω N } ( A ) = χ Ω − ( ω ) χ Ω +1 ( ω ) × (cid:20) ∆ S +1 ( ω n ) V ( ω , ω ) µ ( ω ) { ω ,ω } ,..., { ω N ,ω N } ( A ) + ∆ S − ( ω ) V ( ω , ω ) µ ( ω ) { ω ,ω } ,..., { ω N ,ω N } ( A ) (cid:21) , (113)where we put µ ( ω ,...,ω N − ,ω N ) N ( A ) = χ A ( ω , . . . , ω N − , ω N ) , A ∈ F N . (114)The recurrent relations (111) - (113) we call the recurrent relations for the spotmeasures µ { ω n ,ω n } ,..., { ω N ,ω N } ( A ) . ψ n ( ω , . . . , ω n ) = χ Ω − n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n )+ χ Ω + n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n ) , (115)where ψ n ( ω , . . . , ω n − , ω n ) = χ Ω + n ( ω , . . . , ω n − , ω n ) × ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − , (116) ψ n ( ω , . . . , ω n − , ω n ) = χ Ω − n ( ω , . . . , ω n − , ω n ) × ∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − . (117) Lemma 6.
For the spot measure µ { ω ,ω } ,..., { ω N ,ω N } ( A ) the representation µ { ω ,ω } ,..., { ω N ,ω N } ( A ) = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) χ A ( ω i , . . . , ω i N N ) , A ∈ F N , (118) is true.Proof. The proof of Lemma 6 we lead by induction down. Let us prove the equality µ ( ω ,...,ω N − ) { ω N ,ω N } ( A ) = X i N =1 ψ N ( ω , . . . , ω N − , ω i N N ) χ A ( ω , . . . , ω N − , ω i N N ) . (119)Really, ψ N ( ω , . . . , ω N − , ω N ) χ A ( ω , . . . , ω N − , ω N )+ ψ N ( ω , . . . , ω N − , ω N ) χ A ( ω , . . . , ω N − , ω N ) = (cid:20) χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) + χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) ∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) (cid:21) × χ A ( ω , . . . , ω N − , ω N )+28 χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) + χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) ∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) (cid:21) × χ A ( ω , . . . , ω N − , ω N ) = χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N ) (cid:21) , A ∈ F N . (120)The last prove the needed. Suppose that we proved that the equality µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A ) = X i n +1 =1 . . . X i N =1 N Y j = n +1 ψ j ( ω , . . . , ω n , ω i n +1 n +1 , . . . , ω i j j ) χ A ( ω , . . . , ω n , ω i n +1 n +1 , . . . , ω i N N ) ,A ∈ F N , (121)is true. By the same way as above, we have X i n =1 ψ n ( ω , . . . , ω n − , ω i n n ) µ ( ω ,...,ω n − ,ω inn ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A ) = χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A ) (cid:21) = µ ( ω ,...,ω n − ) { ω n ,ω n } ,..., { ω N ,ω N } ( A ) , A ∈ F N . (122)The last proves Lemma 6. 29et us define the integral for the random value f N ( ω , . . . , ω N − , ω N ) relative tothe measure µ { ω ,ω } ,..., { ω N ,ω N } ( A ) by the formula Z Ω N f N ( ω , . . . , ω N − , ω N ) dµ { ω ,ω } ,..., { ω N ,ω N } = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) f N ( ω i , . . . , ω i N N ) . (123)To describe the convex set of equivalent martingale measures, we introduce thefamily of α -spot measures, depending on the point ( { ω , { ω } , . . . , { ω N , { ω N } ) be-longing to Ω N × Ω N and the set of strictly positive random values α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) , n = 1 , N , (124)at points W n = ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , being constructed by the point( { ω , ω } , . . . , { ω N , ω N } ) . Further, in this section, we assume that the evolution of risky asset is given bythe formula (9). Therefore, in this caseΩ − n = Ω − n × Ω n − , Ω + n = Ω n × Ω n − , n = 1 , N , (125)and the condition (16) is formulated, as follows: Z Ω n × Ω n χ Ω − n ( ω n ) χ Ω n ( ω n ) α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × dP n ( ω n ) dP n ( ω n ) = 1 , n = 1 , N . (126)Let us determine the random values ψ αn ( ω , . . . , ω n ) = χ Ω − n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n )+ χ Ω + n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n ) , (127) ψ n ( ω , . . . , ω n − , ω n ) = α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) χ Ω + n ( ω , . . . , ω n − , ω n ) × ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − , (128) ψ n ( ω , . . . , ω n − , ω n ) = α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) χ Ω − n ( ω , . . . , ω n − , ω n ) × ∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − . (129)30et us define the set of α -spot measures on the σ -algebra F N by the formula µ αW N ( A ) = X i =1 . . . X i N =1 N Y j =1 ψ αj ( ω i , . . . , ω i j j ) χ A ( ω i , . . . , ω i N N ) , A ∈ F N , (130)and the set of the measures µ ( A ) = Z Ω N × Ω N X i =1 . . . X i N =1 N Y j =1 ψ αj ( ω i , . . . , ω i j j ) χ A ( ω i , . . . , ω i N N ) dP N × dP N , A ∈ F N . (131) Theorem 6.
Suppose that the conditions of Lemma 1 are true. If the strictly positiverandom values α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) , n = 1 , N , (132) given on the probability space { Ω n × Ω n , F n × F n , P n × P n } , n = 1 , N , satisfy theconditions (126), then for the measure µ ( A ) , given by the formula (131), the rep-resentation µ ( A ) = Z Ω N × Ω N N Y i =1 α i ( { ω , . . . , ω i } ; { ω , . . . , ω i } ) µ { ω ,ω } ,..., { ω N ,ω N } ( A ) dP N × dP N (133) is true.Proof. Due to Lemma 1, the set of random values α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) ,n = 1 , N , satisfying the conditions (126), is a non empty set.We prove Theorem 6 by induction down. For the spot measure the relation µ ( ω ,...,ω N − ) { ω N ,ω N } ( A ) = χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N ) (cid:21) , A ∈ F N , (134)is true. Multiplying the relation (134) on α N ( { ω , . . . , ω N − , ω N } ; { ω , . . . , ω N − , ω N } )and after that, integrating relative to the measure P N × P N on the set Ω N × Ω N , weobtain Z Ω N Z Ω N α N ( { ω , . . . , ω N − , ω N } ; { ω , . . . , ω N − , ω N } ) × µ ( ω ,...,ω N − ) { ω N ,ω N } ( A ) dP N ( ω N ) dP N ( ω N ) =31 Ω N Z Ω N α N ( { ω , . . . , ω N − , ω N } ; { ω , . . . , ω N − , ω N } ) × χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N ) (cid:21) dP N ( ω N ) dP N ( ω N ) = µ ( ω ,...,ω N − ) N − ( A ) , A ∈ F N . (135)Suppose that we proved the equality Z N Q i = n +1 [Ω i × Ω i ] N Y i = n +1 α i ( { ω , . . . , ω n , ω n +1 , . . . , ω i } ; { ω , . . . , ω n , ω n +1 , . . . , ω i } ) × µ ( ω ,...,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A ) N Y i = n +1 dP i ( ω i ) dP i ( ω i ) = µ ( ω ,...,ω n ) n ( A ) . (136)Then, using the induction supposition (136), the relation for the spot measure µ ( ω ,...,ω n − ) { ω n ,ω n } ,..., { ω N ,ω N } ( A ) = χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) { ω n +1 ,ω n +1 } ,..., { ω N ,ω N } ( A ) (cid:21) , A ∈ F N , (137)and multiplying it on N Q i = n α i ( { ω , . . . , ω n − , ω n , . . . , ω i } ; { ω , . . . , ω n − , ω n , . . . , ω i } )and then integrating relative to the measure N Q i = n [ P i × P i ] on the set N Q i = n [Ω i × Ω i ] , we obtain the equality Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) (cid:21) dP n ( ω n ) dP n ( ω n ) = µ ( ω ,...,ω n − ) n − ( A ) , n = 1 , N . (138)Thus, we proved the following recurrent relations µ ( ω ,...,ω n − ) n − ( A ) = Z Ω n × Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) (cid:21) dP n ( ω n ) dP n ( ω n ) , n = 1 , N . (139)To finish the proof of Theorem 6, let us calculate Z Ω N × Ω N X i N =1 ψ αN ( ω , . . . , ω N − , ω i N N ) χ A ( ω , . . . , ω N − , ω i N N ) dP N ( ω N ) dP N ( ω N ) . (140)Calculating the expression X i N =1 ψ αN ( ω , . . . , ω N − , ω i N N ) χ A ( ω , . . . , ω N − , ω i N N ) = ψ αN ( ω , . . . , ω N − , ω N ) χ A ( ω , . . . , ω N − , ω N )+ ψ αN ( ω , . . . , ω N − , ω N ) χ A ( ω , . . . , ω N − , ω N ) = α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) × χ Ω − N ( ω , . . . , ω N − , ω N ) χ Ω + N ( ω , . . . , ω N − , ω N ) × (cid:20) ∆ S + N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N )+∆ S − N ( ω , . . . , ω N − , ω N ) V N ( ω , . . . , ω N − , ω N , ω N ) χ A ( ω , . . . , ω N − , ω N ) (cid:21) , A ∈ F N , (141)and substituting (141) into (140), we obtain the equality Z Ω N × Ω N X i N =1 ψ αN ( ω , . . . , ω N − , ω i N N ) χ A ( ω , . . . , ω N − , ω i N N ) dP N ( ω N ) dP N ( ω N ) =33 ( ω ,...,ω N − ) N − ( A ) . (142)Suppose that we already proved the equality Z N Q i = n +1 Ω i × Ω i X i n +1 =1 . . . X i N =1 N Y j =1 ψ αj ( ω , . . . , ω n , ω i n +1 n +1 . . . , ω i j j ) N Y i = n +1 dP i ( ω i ) dP i ( ω i ) = µ ( ω ,...,ω n ) n ( A ) . (143)Then, acting as above, we obtain the equalities Z Ω n × Ω n X i n =1 ψ αn ( ω , . . . , ω n − , ω i n n ) µ ( ω ,...,ω n − ,ω inn ) n ( A ) dP n ( ω n ) dP n ( ω n ) = Z Ω n × Ω n α n ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) × χ Ω − n ( ω , . . . , ω n − , ω n ) χ Ω + n ( ω , . . . , ω n − , ω n ) × (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A )+∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) µ ( ω ,...,ω n − ,ω n ) n ( A ) (cid:21) dP n ( ω n ) dP n ( ω n ) = µ ( ω ,...,ω n − ) n − ( A ) , A ∈ F N . (144)We proved that the recurrent relations (144) are the same as the recurrent relations(139). This proves Theorem 6.Let us introduce the denotations µ { ω ,ω } ,..., { ω N ,ω N } (Ω N ) = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) ,W N = { ω , . . . , ω N ; ω , . . . , ω N } = {{ ω } N , { ω } N } . (145)Further, only those points ( { ω , ω } , . . . , { ω N , ω N } ) ∈ Ω N × Ω N play important rolefor which µ { ω ,ω } ,..., { ω N ,ω N } (Ω N ) = 0 . Below, in the next two Theorems, we assume that the random value α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) (146)given on the probability space { Ω n × Ω n , F n × F n , P n × P n } , n = 1 , N , satisfy theconditions (126). 34nder the above conditions, for the measure µ ( A ) , given by the formula (133),the representation µ ( A ) = Z Ω N N Y n =1 ψ n ( ω , . . . , ω n ) χ A ( ω , . . . , ω N ) N Y i =1 dP i ( ω i ) (147)is true, where ψ n ( ω , . . . , ω n ) = χ Ω − n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n )+ χ Ω + n ( ω , . . . , ω n − , ω n ) ψ n ( ω , . . . , ω n ) , (148) ψ n ( ω , . . . , ω n − , ω n ) = Z Ω n χ Ω + n ( ω , . . . , ω n − , ω n ) α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − , (149) ψ n ( ω , . . . , ω n − , ω n ) = Z Ω n χ Ω − n ( ω , . . . , ω n − , ω n ) α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) × ∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) , ( ω , . . . , ω n − ) ∈ Ω n − . (150)Due to the conditions (126) relative to the random values α n ( { ω } n ; { ω } n ) , we have Z Ω n ψ n ( ω , . . . , ω n ) dP n ( ω n ) = 1 , n = 1 , N . (151)for ψ n ( ω , . . . , ω n ) , given by the formula (148). The proof of the equalities (151) isthe same as in Theorem 1. Theorem 7.
Suppose that the conditions of Lemma 1 are true. Then, the set ofstrictly positive random values α n ( { ω } n ; { ω } n ) , n = 1 , N , satisfying the conditions E µ | ∆ S n ( ω , . . . , ω n − , ω n ) | = Z Ω N N Y i =1 ψ i ( ω , . . . , ω i ) | ∆ S n ( ω , . . . , ω n − , ω n ) | N Y i =1 dP i ( ω i ) < ∞ , n = 1 , N , (152) is a non empty set for the measures µ ( A ) , given by the formula (133). The measure µ ( A ) , constructed by the strictly positive random values α n ( { ω } n ; { ω } n ) , n = 1 , N , satisfying the conditions (126), (152) is a martingale measure for the evolution ofrisky asset, given by the formula (9). Every measure, belonging to the convex linearspan of such measures, is also martingale measure for the evolution of risky asset,given by the formula (9). They are equivalent to the measure P N . The set of spotmeasures µ { ω ,ω } ,..., { ω N ,ω N } ( A ) is a set of martingale measures for the evolution ofrisky asset, given by the formula (9). roof. The first fact, that the set of random values α n ( { ω } n ; { ω } n ) , n = 1 , N , satis-fying the conditions (126), (152) is a non empty one, follows from Lemma 1. Fromthe representation (147) for the set of measures µ ( A ), given by the formula (133), asin the proof of Theorem 1, it is proved that this set of measures is a set of martingalemeasures being equivalent to the measure P N . Let us prove the last statement of Theorem 7. Since for the spot measure µ { ω ,ω } ,..., { ω N ,ω N } ( A ) the representation µ { ω ,ω } ,..., { ω N ,ω N } ( A ) = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) χ A ( ω i , . . . , ω i N N ) , A ∈ F N , (153)is true, let us calculate X i j =1 ψ j ( ω i , . . . , ω i j j ) = ψ j ( ω i , . . . , ω i j − j − , ω j ) + ψ j ( ω i , . . . , ω i j − j − , ω j ) = χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) ψ j ( ω i , . . . , ω i j − j − ω j )+ χ Ω + n ( ω i , . . . , ω i j − j − , ω j ) ψ j ( ω i , . . . , ω i j − j − ω j )+ χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) ψ j ( ω i , . . . , ω i j − j − ω j )+ χ Ω + n ( ω i , . . . , ω i j − j − , ω j ) ψ j ( ω i , . . . , ω i j − j − ω j ) = χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) ∆ S + j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) + χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) ∆ S − j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) + χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) ∆ S + j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) + χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) ∆ S − j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) = χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) ∆ S + j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) +36 Ω + j ( ω i , . . . , ω i j − j − , ω j ) χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) ∆ S − j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) = χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) = χ Ω − j ( ω j ) χ Ω j ( ω j ) = (cid:26) , ω j ∈ Ω − j ω j ∈ Ω j , , otherwise, , j = 1 , N . (154)Further, X i j =1 ψ j ( ω i , . . . , ω i j j )∆ S j ( ω i , . . . , ω i j j ) = ψ j ( ω i , . . . , ω i j − j − , ω j )∆ S j ( ω i , . . . , ω i j − j − , ω j )+ ψ j ( ω i , . . . , ω i j − j − , ω j )∆ S j ( ω i , . . . , ω i j − j − , ω j ) = χ Ω − j ( ω i , . . . , ω i j − j − , ω j ) χ Ω + j ( ω i , . . . , ω i j − j − , ω j ) × " − ∆ S + j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) ∆ S − j ( ω i , . . . , ω i j − j − , ω j )+∆ S − j ( ω i , . . . , ω i j − j − , ω j ) V j ( ω i , . . . , ω i j − j − , ω j , ω j ) ∆ S + j ( ω i , . . . , ω i j − j − , ω j ) = 0 , j = 1 , N . (155)Let us prove that the set of measures µ { ω ,ω } ,..., { ω N ,ω N } ( A ) is a set of martingalemeasures. Really, for A, belonging to the σ -algebra F n − of the filtration we have A = B × N Q i = n Ω i , where B belongs to σ -algebra F n − of the measurable space { Ω n − , F n − } . Then, Z A ∆ S n ( ω , . . . , ω n ) dµ { ω ,ω } ,..., { ω N ,ω N } = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) χ B ( ω i , . . . , ω i n − n − )∆ S n ( ω i , . . . , ω i n n ) = X i =1 . . . X i n =1 n Y j =1 ψ j ( ω i , . . . , ω i j j ) χ B ( ω i , . . . , ω i n − n − )∆ S n ( ω i , . . . , ω i n n ) = X i =1 . . . X i n − =1 n − Y j =1 ψ j ( ω i , . . . , ω i j j ) χ B ( ω i , . . . , ω i n − n − ) × X i n =1 ψ n ( ω i , . . . , ω i n n )∆ S n ( ω i , . . . , ω i n n ) = 0 , A ∈ F n − . (156)The last means the needed statement. Theorem 7 is proved.Below, in Theorem 8, we present the consequence of Theorems 6, 7. Theorem 8.
Let the evolution of risky asset be given by the formula (9) and letLemma 1 conditions be true. Suppose that the random value α N ( { ω } N ; { ω } N ) , givenon the probability space { Ω − N × Ω + N , F − N × F + N , P − N × P + N } , satisfy the conditions P − N × P + N (( { ω , . . . , ω N } ; { ω , . . . , ω N } ) , α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) >
0) = N Y n =1 P n (Ω − n ) × P n (Ω n ); (157) Z Ω − n × Ω n α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) × ∆ S + n ( ω , . . . , ω n − , ω n )∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) dP n ( ω n ) dP n ( ω n ) < ∞ , ( ω , . . . , ω n − ) ∈ Ω n − ; (158) Z N Q i =1 [Ω − i × Ω i ] α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) N Y i =1 dP i ( ω i ) dP i ( ω i ) = 1 , (159) where α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) = (160) R N Q i = n +1 [Ω − i × Ω i ] α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) N Q i = n +1 dP i ( ω i ) dP i ( ω i ) R N Q i = n [Ω − i × Ω i ] α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) N Q i = n dP i ( ω i ) dP i ( ω i ) , n = 1 , N . If the set of strictly positive random values α n ( { ω } n ; { ω } n ) , n = 1 , N , given bythe formula (160), satisfies the condition E µ | ∆ S n ( ω , . . . , ω n − , ω n ) | =38 Ω N N Y i =1 ψ i ( ω , . . . , ω i ) | ∆ S n ( ω , . . . , ω n − , ω n ) | N Y i =1 dP i ( ω i ) < ∞ , n = 1 , N , (161) then, for the martingale measure µ ( A ) the representation µ ( A ) = Z Ω N × Ω N α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) µ { ω ,ω } ,..., { ω N ,ω N } ( A ) dP N × dP N (162) is true.Proof. The random values α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) , n = 1 , N , sat-isfy the conditions (14) - (16), due to the conditions of Theorem 8. It is evidentthat α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) = N Y n =1 α n ( { ω , . . . , ω n } ; { ω , . . . , ω n } ) . (163)Due to Theorem 7, µ ( A ) , given by the formula (162), is a martingale measure beingequivalent to the measure P N . Let us indicate how to construct the random values α N ( { ω } N ; { ω } N ) , since theserandom values determine the set of all martingale measures. Suppose that therandom value α ki ( ω i , ω i ) , k = 1 , K, is a bounded strictly positive random value,given on the measurable space { Ω − i × Ω i , F − i × F i } , i = 1 , N , and satisfyingthe conditions Z Ω − i × Ω i α ki ( ω i , ω i ) dP i ( ω i ) dP i ( ω i ) = 1 , i = 1 , N , k = 1 , K. (164)Let us denote α kN ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) = N Y i =1 α ki ( ω i , ω i ) , k = 1 , K, (165)where K runs natural numbers. If γ k , k = 1 , K, are strictly positive real numberssuch that K P k =1 γ k = 1 , then α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) = K X k =1 γ k α kN ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) (166)satisfy the conditions of Theorem 8. The set of random values (166) is dense in theset of random values α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) , satisfying the condition (157)- (159). Theorem 8 is proved. 39nother way to construct α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) is to use the equalities(126). The set of α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) can construct as follows:suppose that α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) satisfies the inequalities0 < h n ≤ α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) ≤ H n < ∞ (167)for a certain real positive numbers h n , H n . If to put α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) = α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) R Ω − n × Ω n α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) dP n ( ω n ) dP n ( ω n ) , (168)then the set of random values α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) , n = 1 , N , is bounded and satisfy the conditions (14) - (16) under the conditions of Theorem7. We can put α N ( { ω , . . . , ω N } ; { ω , . . . , ω N } ) = N Y n =1 α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) . (169)It is evident that α n ( { ω , . . . , ω n − , ω n } ; { ω , . . . , ω n − , ω n } ) , n = 1 , N , must satisfythe conditions (161). In the papers [27], [28], the range of non arbitrage prices are established. In thepaper [27], for the Levy exponential model, the price of super-hedge for call optioncoincides with the price of the underlying asset under the assumption that theLevy process has unlimited variation, does not contain a Brownian component,with negative jumps of arbitrary magnitude. The same result is true, obtained inthe paper [28], if the process describing the evolution of the underlying asset is adiffusion process with the jumps described by Poisson jump process. In these papersthe evolution is described by continuous processes. Below, we consider the discreteevolution of risky assets that is more realistic from the practical point of view. Twotypes of risky asset evolutions are considered: 1) the price of an asset can take anynon negative value; 2) the price of the risky asset may not fall below a given positivevalue for finite time of evolution. For each of these types of evolutions of risky assets,the bounds of non-arbitrage prices for a wide class of contingent liabilities are found,among which are the payoff functions of standard call and put options.Below, on the probability space { Ω N , F N , P N } , where Ω N = N Q i =1 Ω i , F N = N Q i =1 F i ,P N = N Q i =1 P i , Ω i is a complete separable metric space, F i is a Borel σ -algebra onΩ i , P i is a probability measure on F i , i = 1 , N , we consider the evolution of riskyasset given by the formula S n ( ω , . . . , ω n ) =40 n Y i =1 (1 + a i ( ω , . . . , ω i − )( e σ i ( ω ,...,ω i − ) ε i ( ω i ) − , n = 1 , N , (170)where a i ( ω , . . . , ω i − ) , σ i ( ω , . . . , ω i − ) are F i − -measurable random values, satis-fying the conditions 0 < a i ( ω , . . . , ω i − ) ≤ , σ i ( ω , . . . , ω i − ) ≥ σ i > , where σ i , i = 1 , N , are real positive numbers. Further, we assume that the random value ε i ( ω i ) satisfy the conditions: there exists ω i ∈ Ω i such that ε i ( ω i ) = 0 , i = 1 , N , and for every real number t > , P i ( ε i ( ω i ) < − t ) > , P i ( ε i ( ω i ) > t ) > , i = 1 , N . For the evolution of risky asset (170), we have∆ S n ( ω , . . . , ω n − , ω n ) = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − )( e σ n ( ω ,...,ω n − ) ε n ( ω n ) −
1) = (171) d n ( ω , . . . , ω n − , ω n )( e σ n ε n ( ω n ) − , where d n ( ω , . . . , ω n − , ω n ) = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − ) ( e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ε n ( ω n ) − . (172)It is evident that d n ( ω , . . . , ω n − , ω n ) > , therefore for ∆ S n ( ω , . . . , ω n − , ω n ) therepresentation (60) is true with η n ( ω n ) = ( e σ n ε n ( ω n ) − . Therefore,∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) = e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) , ω n ∈ Ω n , ( ω , . . . , ω n − ) ∈ Ω n − , (173)∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) =1 − e σ n ( ω ,...,ω n − ) ε n ( ω n ) e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) , ω n ∈ Ω − n , ( ω , . . . , ω n − ) ∈ Ω n − , (174)where we denotedΩ − n = { ω n ∈ Ω n , ε n ( ω n ) ≤ } , Ω n = { ω n ∈ Ω n , ε n ( ω n ) > } , Ω − n = Ω − n × Ω n − , Ω + n = Ω n × Ω n − . (175)From the formulas (173), (174) and Theorem 1, it follows that the set of martin-gale measures M do not depend on the random values a i ( ω , . . . , ω i − ) , i = 1 , N . If41o put a i ( ω , . . . , ω i − ) = 1 , i = 1 , N , in the formula (170), then for the risky assetevolution we obtain the formula S n ( ω , . . . , ω n − , ω n ) = S n Y i =1 e σ i ( ω ,...,ω i − ) ε i ( ω i ) , n = 1 , N . (176)The evolution of risky assets, given by the formula (176), includes a wide classof evolutions of risky assets, used in economics. For example, under an appro-priate choice of probability spaces { Ω i , F i , P i } and a choice of sequence of in-dependent random values ε i ( ω i ) with the normal distribution N (0 , , we obtainARCH model (Autoregressive Conditional Heteroskedastic Model) introduced byEngle in [18] and GARCH model (Generalized Autoregressive Conditional Het-eroskedastic Model) introduced later by Bollerslev in [19]. In these models, therandom variables σ i ( ω , . . . , ω i − ) , i = 1 , N , are called the volatilities which satisfythe nonlinear set of equations.The very important case of evolution of risky assets (170) is when a i ( ω , . . . , ω i − ) = a i , i = 1 , N , are constants, that is, S n ( ω , . . . , ω n − , ω n ) = S n Y i =1 (1 + a i ( e σ i ( ω ,...,ω i − ) ε i ( ω i ) − , n = 1 , N , (177)where 0 ≤ a i ≤ . If 0 < a i < , i = 1 , N , then the evolution of risky asset, given by the formula(177), we call the evolution of relatively stable asset.Further, we assume that the evolution of risky asset given by the formulas (170),(176), (177) satisfy the conditions Z Ω N S n ( ω , . . . , ω n − , ω n ) dP N < ∞ , n = 1 , N . (178)From the conditions (178), it follows the inequalities Z Ω N ∆ S − n ( ω , . . . , ω n ) dP N < ∞ , n = 1 , N . (179)Taking into account that ∆ S − n ( ω , . . . , ω n − , ω n ) = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − )(1 − e σ n ( ω ,...,ω n − ) ε n ( ω n ) ) , ω n ∈ Ω − n , (180)∆ S + n ( ω , . . . , ω n − , ω n ) = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − )( e σ n ( ω ,...,ω n − ) ε n ( ω n ) − , ω n ∈ Ω n , (181)we have 1∆ S − n ( ω , . . . , ω n − , ω n ) ≤ n − Q i =1 (1 − a i ) a n (1 − e σ n ε n ( ω n ) ) < ∞ , ε n ( ω n ) < , (182)42∆ S + n ( ω , . . . , ω n − , ω n ) ≤ n − Q i =1 (1 − a i ) a n ( e σ n ε n ( ω n ) − < ∞ , ε n ( ω n ) > , (183)under the conditions that0 < a n ≤ a n ( ω , . . . , ω n − ) ≤ a n < , n = 1 , N . (184) Theorem 9.
On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by one of the formula (170), (176), (177) that satisfies the conditions (178).If the inequalities < a n ≤ a n ( ω , . . . , ω n − ) ≤ a n < , < a i < , i = 1 , N , aretrue, then the set of martingale measures M is the same for every evolution of riskyassets, given by the formulas (170), (177). For every non-negative super-martingalerelative to the set of martingale measures M the optional decomposition is valid.Every measure of M is an integral over the spot measures. The fair price f ofsuper-hedge for the nonnegative payoff function f ( x ) is given by the formula f = sup P ∈ M E P f ( S N ) = sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } . (185) The set of martingale measures M for the evolution of risky asset, given by theformula (176), is contained in the set M. Proof.
From the equalities (173) - (174) and the inequalities (178), it follows thatthe set M is a nonempty one and every martingale measure constructed by the setof random values α n ( ω , . . . , ω n ; ω , . . . , ω n ) , n = 1 , N , belongs to the set M, if theinequalities (49) are true. To prove that the set of martingale measures, defined bythe evolutions (170), (177), coincide it is necessary to prove the inequalities0 < A n ≤ S n ( ω , . . . , ω n ) S n ( ω , . . . , ω n ) ≤ B n < ∞ , n = 1 , N , (186)where we denoted by S n ( ω , . . . , ω n ) the evolution, given by the formula (170), andby S n ( ω , . . . , ω n ) the evolution, given by the formula (177). Under the conditionsof Theorem 9, we have S n ( ω , . . . , ω n ) S n ( ω , . . . , ω n ) = S n Q i =1 (1 + a i ( ω , . . . , ω i − )( e σ i ( ω ,...,ω i − ) ε i ( ω i ) − S n Q i =1 (1 + a i ( e σ i ( ω ,...,ω i − ) ε i ( ω i ) − , n = 1 , N . (187)Since 1 + a i ( ω , . . . , ω i − )( e σ i ( ω ,...,ω i − ) ε i ( ω i ) − a i ( e σ i ( ω ,...,ω i − ) ε i ( ω i ) −
1) =1 − a i ( ω , . . . , ω i − ) + a i ( ω , . . . , ω i − ) e σ i ( ω ,...,ω i − ) ε i ( ω i ) − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) = D i , i = 1 , N , (188)43e have 1 − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) ≤ D i ≤ − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) , i = 1 , N . (189)Let us denote A i = inf ( ω ,...,ω i ) ∈ Ω i − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) , i = 1 , N ,B i = sup ( ω ,...,ω i ) ∈ Ω i − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) − a i + a i e σ i ( ω ,...,ω i − ) ε i ( ω i ) , i = 1 , N . (190)It is evident that 0 < A i , B i < ∞ , i = 1 , N , and A i ≤ D i ≤ B i , i = 1 , N , (191)therefore A n = n Y i =1 A i ≤ S n ( ω , . . . , ω n ) S n ( ω , . . . , ω n ) ≤ n Y i =1 B i = B n , n = 1 , N . (192)So, A N ≤ S n ( ω , . . . , ω n ) S n ( ω , . . . , ω n ) ≤ B N , n = 1 , N , (193)where we put A N = min ≤ n ≤ N A n , B N = max ≤ n ≤ N B n . Since | ∆ S n ( ω , . . . , ω n − , ω n ) | = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − ) | ( e σ n ( ω ,...,ω n − ) ε n ( ω n ) − | , (194) | ∆ S n ( ω , . . . , ω n − , ω n ) | = S n − ( ω , . . . , ω n − ) a n | ( e σ n ( ω ,...,ω n − ) ε n ( ω n ) − | , (195)we have | ∆ S n ( ω , . . . , ω n − , ω n ) || ∆ S n ( ω , . . . , ω n − , ω n ) | = S n − ( ω , . . . , ω n − ) a n ( ω , . . . , ω n − ) S n − ( ω , . . . , ω n − ) a n . (196)44aking into account the obtained inequalities, we have the inequalities A N min ≤ n ≤ N a n max ≤ n ≤ N a n ≤ | ∆ S n ( ω , . . . , ω n − , ω n ) || ∆ S n ( ω , . . . , ω n − , ω n ) | ≤ B N max ≤ n ≤ N a n min ≤ n ≤ N a n , n = 1 , N . (197)The inequalities (197) proves that the set of martingale measures for the evolutionsof risky assets given by the formulas (170), (177) are the same, since the inequalities(49) for the evolutions of risky assets, given by formulas (170), (177), are fulfilledsimultaneously.For the evolution of risky assets (177), satisfying the conditions (184), the in-equalities (182), (183) are true. From this, it follows that the conditions of Theorem5 are valid. This proves the optional decomposition for every nonnegative super-martingale relative to the family of martingale measures M. From [17], it followsthe formula for the fair price f of super-hedge f = sup P ∈ M E P f ( S N ) . (198)Further, the conditions of Theorem 8 is also true. Therefore, the formulasup P ∈ M E P f ( S N ) = sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } (199)is valid.To complete the proof of Theorem 9, it needs to show that the set M ⊆ M. Letus denote S n ( ω , . . . , ω n ) the evolution of risky asset, given by the formula (176).Then, as above S n ( ω , . . . , ω n ) S n ( ω , . . . , ω n ) ≤ n Y i =1 a i = C n , n = 1 , N . (200)Therefore, | ∆ S n ( ω , . . . , ω n − , ω n ) || ∆ S n ( ω , . . . , ω n − , ω n ) | = S n − ( ω , . . . , ω n − ) S n − ( ω , . . . , ω n − ) a n ≤ max ≤ n ≤ N C n min ≤ n ≤ N a n , n = 1 , N . (201)The inequality (201) proves the needed statement. Theorem 9 is proved. Theorem 10.
On the probability space { Ω N , F N , P N } , let the evolution of riskyasset be given by the formula (170). Suppose that ≤ a i ( ω , . . . , ω i − ) ≤ ,σ i ( ω , . . . , ω i − ) > σ i > , i = 1 , N , and a n = 1 for a certain ≤ n ≤ N. Ifthe nonnegative payoff function f ( x ) , x ∈ [0 , ∞ ) , satisfies the conditions:1) f (0) = 0 , f ( x ) ≤ ax, lim x →∞ f ( x ) x = a, a > , then sup P ∈ M E P f ( S N ) = aS . (202)45 f, in addition, the nonnegative payoff function f ( x ) is a convex down one, then inf P ∈ M E P f ( S N ) = f ( S ) , (203) where M is a set of equivalent martingale measures for the evolution of risky as-set, given by the formula (170). The interval of non-arbitrage prices of contingentliability f ( S N ) lies in the set [ f ( S ) , aS ] . Proof.
Since the conditions of Theorem 9 are satisfied, then the formulasup Q ∈ M Z Ω N f ( S N ) dQ = sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } (204)is true, where for the spot measure µ { ω ,ω } ,..., { ω N ,ω N } ( A ) the representation µ { ω ,ω } ,..., { ω N ,ω N } ( A ) = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) χ A ( ω i , . . . , ω i N N ) , A ∈ F N , (205)is valid, and sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } =sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! , (206)where we denoted Ω − s = { ω s ∈ Ω s , ε s ( ω s ) ≤ } , Ω s = { ω s ∈ Ω s , ε s ( ω s ) > } . From the inequality, f ( S N ) ≤ aS N , we havesup Q ∈ M Z Ω f ( S N ) dQ ≤ aS . (207)To prove the inverse inequality, we use the inequalitysup Q ∈ M Z Ω f ( S N ) dQ ≥ X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! . (208)46n the right hand side of the last inequality, let us put ε s ( ω s ) = 0 , s = n. Suchelementary events ω s exist, due to the conditions relative to the random values ε s ( ω s ) , s = 1 , N . We obtain X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! = X i n =1 ψ n ( ω , . . . , ω n − , ω i n n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω inn ) (cid:17) . (209)Therefore, sup Q ∈ M Z Ω f ( S N ) dQ ≥ sup ω n ∈ Ω − n ,ω n ∈ Ω n X i n =1 ψ n ( ω , . . . , ω n − , ω i n n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω inn ) (cid:17) . (210)Further, sup ω n ∈ Ω − n ,ω n ∈ Ω n X i n =1 ψ n ( ω , . . . , ω i n n ) × f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω inn ) (cid:17) =sup ω n ∈ Ω − n ,ω n ∈ Ω n (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) +∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17)(cid:21) ≥ lim ε n ( ω n ) →∞ lim ε n ( ω n ) →−∞ " e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) × f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) +1 − e σ n ( ω ,...,ω n − ) ε n ( ω n ) e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) =lim ε n ( ω n ) →∞ e σ n ( ω ,...,ω n − ) ε n ( ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) = aS . (211)47ubstituting the inequality (211) into the inequality (209), we obtain the neededinequality.Let us prove the equality (203). Using the Jensen inequality, we obtaininf P ∈ M E P f ( S N ) ≥ f ( E P S N ) = f ( S ) . (212)Let us prove the inverse inequality. It is evident that X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ inf P ∈ M E P f ( S N ) . (213)Putting in this inequality ε i ( ω i ) = 0 , i = 1 , N , we obtain the needed. The last state-ment about the interval of non-arbitrage prices follows from [7] and [6]. Theorem10 is proved. Theorem 11.
On the probability space { Ω N , F N , P N } , let the evolution of riskyasset be given by the formula (170). Suppose that ≤ a i ( ω , . . . , ω i − ) ≤ ,σ i ( ω , . . . , ω i − ) > σ i > , i = 1 , N , and a n = 1 for a certain ≤ n ≤ N. Ifthe nonnegative payoff function f ( x ) , x ∈ [0 , ∞ ) , satisfies the conditions:1) f (0) = K, f ( x ) ≤ K, then sup P ∈ M E P f ( S N ) = K. (214) If, in addition, the nonnegative payoff function f ( x ) is a convex down one, then inf P ∈ M E P f ( S N ) = f ( S ) , (215) where M is a set of equivalent maqtingale measures for the evolution of risky as-set, given by the formula (170). The interval of non-arbitrage prices of contingentliability f ( S N ) coincides with the set [ f ( S ) , K ] . Proof.
Due to Theorem 9, the equalitysup Q ∈ M Z Ω N f ( S N ) dQ = sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } (216)is valid, where for the spot measure µ { ω ,ω } ,..., { ω N ,ω N } ( A ) the representation µ { ω ,ω } ,..., { ω N ,ω N } ( A ) = X i =1 . . . X i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) χ A ( ω i , . . . , ω i N N ) , A ∈ F N , (217)48s true, and sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } =sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! . (218)It is evident that sup P ∈ M E P f ( S N ) ≤ K. (219)Further, sup Q ∈ M Z Ω f ( S N ) dQ ≥ X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! . (220)In the right hand side of the last inequality, let us put ε s ( ω s ) = 0 , s = n. We obtain X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! = X i n =1 ψ n ( ω , . . . , ω n − , ω i n n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω inn ) (cid:17) . (221)From the last equality, we obtainsup Q ∈ M Z Ω f ( S N ) dQ ≥ sup ω n ∈ Ω − n ,ω n ∈ Ω n X i n =1 ψ n ( ω , . . . , ω n − , ω i n n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω inn ) (cid:17) . (222)49urther, sup ω n ∈ Ω − n ,ω n ∈ Ω n X i n =1 ψ n ( ω , . . . , ω n − , ω i n n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω inn ) (cid:17) =sup ω n ∈ Ω − n ,ω n ∈ Ω n (cid:20) ∆ S + n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) +∆ S − n ( ω , . . . , ω n − , ω n ) V n ( ω , . . . , ω n − , ω n , ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17)(cid:21) ≥ lim ε ( ω n ) →∞ lim ε ( ω n ) →−∞ " e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) +1 − e σ n ( ω ,...,ω n − ) ε n ( ω n ) e σ n ( ω ,...,ω n − ) ε n ( ω n ) − e σ n ( ω ,...,ω n − ) ε n ( ω n ) f (cid:16) S e σ n ( ω ,...,ω n − ) ε n ( ω n ) (cid:17) = f (0) = K. (223)Substituting the inequality (223) into the inequality (221), we obtain the neededinequality.Let us prove the equality (215). Due to the convexity of the payoff function f ( x ) , using the Jensen inequality, we obtaininf P ∈ M E P f ( S N ) ≥ f ( E P S N ) = f ( S ) . (224)Let us prove the inverse inequality. It is evident that X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s ( ω i , . . . , ω i s − s − ) (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ inf P ∈ M E P f ( S N ) . (225)Putting in this inequality ε i ( ω i ) = 0 , i = 1 , N , we obtain the needed. The last state-ment about the interval of non-arbitrage prices follows from [7] and [6]. Theorem11 is proved. Theorem 12.
On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177). Suppose that ≤ a i ≤ , σ i ( ω , . . . , ω i − ) > σ i > ,i = 1 , N . If the nonnegative payoff function f ( x ) , x ∈ [0 , ∞ ) , satisfies the conditions:1) f (0) = 0 , f ( x ) ≤ ax, lim x →∞ f ( x ) x = a, a > , then the inequalities f S N Y i =1 (1 − a i ) ! + aS − N Y i =1 (1 − a i ) ! ≤ sup P ∈ M E P f ( S N ) ≤ aS (226)50 re true. If, in addition, the nonnegative payoff function f ( x ) is a convex down one,then inf P ∈ M E P f ( S N ) = f ( S ) , (227) where M is the set of equivalent martingale measures for the evolution of risky asset,given by the formula (177).Proof. As before, aS ≥ sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } =sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! . (228)sup ω N ∈ Ω − N ,ω N ∈ Ω N X i N =1 ψ N ( ω i , . . . , ω i N N ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! =sup ω N ∈ Ω − N ,ω N ∈ Ω N " ∆ S + N ( ω i , . . . , ω i N − N − , ω N ) V N ( ω i , . . . , ω i N − N − , ω N , ω N ) × f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17) +∆ S − N ( ω i , . . . , ω i N − N − , ω N ) V N ( ω i , . . . , ω i N − N − , ω N , ω N ) f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17) ≥ lim ε N ( ω N ) →∞ lim ε N ( ω N ) →−∞ " e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) × f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17) +1 − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) × f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17)i =51 ( S N − (1 − a N )) + aa N S N − , (229)where we put S N − = S N − Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17) . (230)Substituting the inequality (229) into (228), we obtain the inequalitysup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N − X i =1 ,...,i N − =1 N − Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S (1 − a N ) N − Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! + aa N S . (231)Applying ( N −
1) times the inequality (231), we obtain the inequalitysup Q ∈ M Z Ω f ( S N ) dQ ≥ f ( S N Y i =1 (1 − a i )) + aS N X i =1 a i N Y s = i +1 (1 − a s ) = f S N Y i =1 (1 − a i ) ! + aS − N Y i =1 (1 − a i ) ! . (232)Let us prove the equality (227). Using the Jensen inequality, we obtaininf P ∈ M E P f ( S N ) ≥ f ( S ) . (233)Let us prove the inverse inequality. It is evident that X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ inf P ∈ M E P f ( S N ) . (234)Putting in the inequality (234) ε n ( ω n ) = 0 , n = 1 , N , we obtain the inverse inequal-ity. 52 heorem 13. On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177). Suppose that ≤ a i ≤ , σ i ( ω , . . . , ω i − ) > σ i > ,i = 1 , N . If the nonnegative payoff function f ( x ) , x ∈ [0 , ∞ ) , satisfies the conditions:1) f (0) = K, f ( x ) ≤ K, then f S N Y i =1 (1 − a i ) ! ≤ sup P ∈ M E P f ( S N ) ≤ K. (235) If, in addition, the nonnegative payoff function f ( x ) is a convex down one, then inf P ∈ M E P f ( S N ) = f ( S ) , (236) where M is the set of equivalent martingale measures for the evolution of risky asset,given by the formula (177).Proof. Let us obtain the estimate from below. Really, aK ≥ sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N Z Ω N f ( S N ) dµ { ω ,ω } ,..., { ω N ,ω N } =sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! . (237)Further,sup ω N ∈ Ω − N ,ω N ∈ Ω N X i N =1 ψ N ( ω i , . . . , ω i N N ) f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! =sup ω N ∈ Ω − N ,ω N ∈ Ω N " ∆ S + N ( ω i , . . . , ω i N − N − , ω N ) V N ( ω i , . . . , ω i N − N − , ω N , ω N ) × f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17) +∆ S − N ( ω i , . . . , ω i N − N − , ω N ) V N ( ω i , . . . , ω i N − N − , ω N , ω N ) f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17) ≥ lim ε N ( ω N ) →∞ lim ε N ( ω N ) →−∞ " e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) × f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17) +53 − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) × f (cid:16) S N − (cid:16) a N (cid:16) e σ N ( ω i ,...,ω iN − N − ) ε N ( ω N ) − (cid:17)(cid:17)(cid:17)i = f ( S N − (1 − a N )) , (238)where we put S N − = S N − Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17) . (239)Substituting the inequality (238) into (237), we obtain the inequalitysup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N − X i =1 ,...,i N − =1 N − Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S (1 − a N ) N − Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! . (240)Applying ( N −
1) times the inequality (240), we obtain the inequalitysup Q ∈ M Z Ω f ( S N ) dQ ≥ f ( S N Y i =1 (1 − a i )) . (241)Let us prove the equality (236). Using the Jensen inequality we obtaininf P ∈ M E P f ( S N ) ≥ f ( S ) . (242)Let us prove the inverse inequality. It is evident that X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ inf P ∈ M E P f ( S N ) . (243)Putting in the inequality (243) ε n ( ω n ) = 0 , n = 1 , N , we obtain the inverse inequal-ity. 54 heorem 14. On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177). Suppose that ≤ a i ≤ , σ i ( ω , . . . , ω i − ) > σ i > ,i = 1 , N . For the payoff function f ( x ) = ( x − K ) + , x ∈ (0 , ∞ ) , K > , the fairprice of super-hedge is given by the formula sup Q ∈ M E Q f ( S N ) = ( S − K ) + , if S N Q i =1 (1 − a i )) ≥ K,S (cid:18) − N Q i =1 (1 − a i ) (cid:19) , if S N Q i =1 (1 − a i ) < K. (244) If S N Q i =1 (1 − a i )) ≥ K, then the set of non arbitrage prices coincides with the point ( S − K ) + , in case if S N Q i =1 (1 − a i ) < K the set of non arbitrage prices coincideswith the set (cid:20) ( S − K ) + , S (cid:18) − N Q i =1 (1 − a i ) (cid:19)(cid:21) . Proof.
Let us introduce the denotations I N = X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! , (245) I N = X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! , (246) I N = sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! , (247)where we put f ( x ) = ( K − x ) + . Let us estimate from above the value I N . For thiswe use the equality I N = I N + S − K, (248)55hich follows from the identity: f ( x ) = f ( x ) + x − K, x ≥ . Since f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≤ f S N Y s =1 (1 − a s ) ! , (249)we obtain the inequality I N ≤ S − K + f S N Y s =1 (1 − a s ) ! . (250)From the inequality (250), we have I N ≤ S − K + f S N Y s =1 (1 − a s )) ! = ( S − K ) + , if S N Q i =1 (1 − a i )) ≥ K,S (cid:18) − N Q i =1 (1 − a i ) (cid:19) , if S N Q i =1 (1 − a i ) < K. (251)Due to the inequality (226) of Theorem 12, I N ≥ f S N Y i =1 (1 − a i ) ! + S − N Y i =1 (1 − a i ) ! (252)and the inequality I N ≥ ( S − K ) + , (253)which follows from the Jensen inequality, we have I N ≥ max ( S − K ) + , f S N Y i =1 (1 − a i ) ! + S − N Y i =1 (1 − a i ) !) = ( S − K ) + , if S N Q i =1 (1 − a i )) ≥ K,S (cid:18) − N Q i =1 (1 − a i ) (cid:19) , if S N Q i =1 (1 − a i ) < K. (254)This proves Theorem 14. Theorem 15.
On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177). Suppose that ≤ a i ≤ , σ i ( ω , . . . , ω i − ) > σ i > ,i = 1 , N . For the payoff function f ( x ) = ( K − x ) + , x ∈ (0 , ∞ ) , K > , the fairprice of super-hedge is given by the formula sup Q ∈ M E Q f ( S N ) = f S N Y i =1 (1 − a i ) ! . (255) The set of non arbitrage prices coincides with the interval (cid:20) ( K − S ) + , f (cid:18) S N Q i =1 (1 − a i ) (cid:19)(cid:21) . roof. The inequality I N = X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≤ f S N Y i =1 (1 − a i ) ! (256)is true. Taking into account the inequality (235) of Theorem 13, we prove Theorem15. Theorem 16.
On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177). Suppose that ≤ a i ≤ , σ i ( ω , . . . , ω i − ) > σ i > ,i = 1 , N . For the payoff function f ( S , S , . . . , S N ) = K − N P i =0 S i N +1 + , K > , thefair price of super-hedge is given by the formula sup Q ∈ M E Q f ( S , S , . . . , S N ) = K − S N P i =0 i Q s =1 (1 − a s ) N + 1 + . (257) The set of non arbitrage prices coincides with the interval ( K − S ) + , K − S N P i =0 i Q s =1 (1 − a s ) N +1 + , if K > S N P i =0 i Q s =1 (1 − a s ) N +1 . For K ≤ S N P i =0 i Q s =1 (1 − a s ) N +1 the set of non arbitrage prices coincides with the point . Proof.
Let us denote S n ( ω , . . . , ω n ) = S n Y s =1 (cid:16) a s (cid:16) e σ s ( ω ,...,ω s − ) ε s ( ω s ) − (cid:17)(cid:17) , n = 1 , N ,t N ( ω , . . . , ω N ) = N Y s =1 e σ s ( ω ,...,ω s − ) ε s ( ω s ) − e σ s ( ω ,...,ω s − ) ε s ( ω s ) − e σ s ( ω ,...,ω s − ) ε s ( ω s ) . (258)It is evident that I N = sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! ≥ (259)57im ε s ( ω s )= −∞ , ε s ( ω s ) →∞ ,s =1 ,N f (cid:0) S , S ( ω ) , . . . , S N ( ω , . . . , ω N ) (cid:1) × t N ( ω , . . . , ω N ) = f S , S (1 − a ) , . . . , S N Y s =1 (1 − a s ) ! = K − S N P i =0 i Q s =1 (1 − a s ) N + 1 + . (260)Let us prove the inverse inequality. We have I N ≤ sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S , S (1 − a ) , . . . , S N Y s =1 (1 − a s ) ! = f S , S (1 − a ) , . . . , S N Y s =1 (1 − a s ) ! = K − S N P i =0 N Q s =1 (1 − a s ) N + 1 + . (261)Therefore, I N ≤ K − S N P i =0 i Q s =1 (1 − a s ) N + 1 + . (262)The inequalities (260), (262) prove Theorem 16. Theorem 17.
On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177). Suppose that ≤ a i ≤ , σ i ( ω , . . . , ω i − ) > σ i > ,i = 1 , N . For the payoff function f ( S , S , . . . , S N ) = N P i =0 S i N +1 − K + , K > , thefair price of super-hedge is given by the formula sup Q ∈ M E Q f ( S , S , . . . , S N ) = ( S − K ) + , if S N P i =0 i Q s =1 (1 − a i ) N +1 ≥ K,S − N P i =0 i Q s =1 (1 − a s ) N +1 , if S N P i =0 i Q s =1 (1 − a s ) N +1 < K. (263)58 f S N P i =0 i Q s =1 (1 − a i ) N +1 ≥ K, then the set of non arbitrage prices coincides with the point ( S − K ) + , in case if S N P i =0 i Q s =1 (1 − a s ) N +1 < K the set of non arbitrage prices coincideswith the interval ( S − K ) + , S − N P i =0 i Q s =1 (1 − a s ) N +1 . Proof.
We have sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! = (264)sup ω i ∈ Ω − i ,ω i ∈ Ω i ,i =1 ,N X i =1 ,...,i N =1 N Y j =1 ψ j ( ω i , . . . , ω i j j ) × f S N Y s =1 (cid:16) a s (cid:16) e σ s ( ω i ,...,ω is − s − ) ε s ( ω iss ) − (cid:17)(cid:17)! + S − K = (265)( S − K ) + K − S N P i =0 i Q s =1 (1 − a s ) N + 1 + = ( S − K ) + , if S N P i =0 i Q s =1 (1 − a i ) N +1 ≥ K,S − N P i =0 i Q s =1 (1 − a s ) N +1 , if S N P i =0 i Q s =1 (1 − a s ) N +1 < K. (266)In the formula (265) we introduced the denotation f ( S , S , . . . , S N ) = K − N P i =0 S i N + 1 + . (267)The proof of Theorem 17 follows from the equality (265).59 Estimation of parameters.
Suppose that { g i ( X N ) } Ni =1 is a mapping from the set [0 , N into itself, where X N = { x , . . . , x N } , ≤ x i ≤ , i = 1 , N . If S , S , . . . , S N is a sample of the process(177), let us denote the order statistic S (0) , S (1) , . . . , S ( N ) of this sample. Introducealso the denotation g i ([ S ] N ) = g i (cid:16) S (0) S ( N ) , . . . , S ( N − S ( N ) (cid:17) , i = 1 , N . Theorem 18.
Suppose that S , S , . . . , S N is a sample of the random process (177).Then, for the parameters a , . . . , a N the estimation a = 1 − τ S (0) S g ([ S ] N ) , < τ ≤ ,a i = 1 − g i ([ S ] N ) g i − ([ S ] N ) , i = 2 , N , (268) is valid, if for g N ([ S ] N ) > , [ S ] N ∈ [0 , N , the inequalities g ([ S ] N ) ≥ g ([ S ] N ) ≥ . . . ≥ g N ([ S ] N ) are true. If τ = 0 , then a i = 1 , i = 1 , N . Proof.
The estimation of the parameters a , . . . , a N we do using the representationof random process S n , n = 1 , N . The smallest value of the random variable S n isequal S n Q i =1 (1 − a i ) , n = 1 , N . Let us determine the parameters a i from the relations S N Y i =1 (1 − a i ) = τ g N ([ S ] N ) , . . . , S N − k Y i =1 (1 − a i ) = τ g N − k ([ S ] N ) , . . . ,S N − k − Y i =1 (1 − a i ) = τ g N − k − ([ S ] N ) , . . . , S (1 − a ) = τ g ([ S ] N ) , (269)where τ > . Taking into account the relations (269), we obtain S (1 − a ) = τ g ([ S ] N ) ,τ g N − k − ([ S ] N ) (1 − a N − k ) = τ g N − k ([ S ] N ) , k = 2 , N . (270)Solving the relations (270), we have a = 1 − τS g ([ S ] N ) , a N − k = 1 − g N − k ([ S ] N ) g N − k − ([ S ] N ) , k = 2 , N . (271)It is evident that a N − k ≥ , k = 2 , N . To provide the positiveness of a and theinequalities τ g N − n ([ S ] N ) ≤ S N − n , n = 0 , N − , S ≥ S (0) , meaning that therandom process (177) takes all the values from the sample S n , n = 0 , N , we mustto put τ = τ S (0) , < τ ≤ . It is evident that, if τ = 0 , then a i = 1 , i = 1 , N Theorem 18 is proved. 60 emark 1.
It is evident that a i = 1 , i = N − k, N , < k ≤ N − , a i = 1 − g i ([ S ] N ) g i − ([ S ] N ) , i = 2 , N − k − ,a = 1 − τ S (0) S g ([ S ] N ) , < τ ≤ , (272) is also estimation of the parameters a , . . . , a N if < g N − k − ([ S ] N ) ≤ g N − k − ([ S ] N ) . . . ≤ g ([ S ] N ) , [ S ] N ∈ [0 , N . Such estimation is not interesting since N − i Y i =1 (1 − a i ) = 0 , i = 0 , k. Remark 2. If g ( x ) = ( S S (0) x, if ≤ x ≤ S (0) S , , if S (0) S < x ≤ , (273) g i ([ S ] N ) = g (cid:18) S ( N − i ) S ( N ) (cid:19) , i = 1 , N , τ = 1 , then for the parameters a , . . . , a N the estimation a i = − S ( N − i ) S ( N − i +1) , if S ( N − i +1) S ( N ) ≤ S (0) S , − S ( N − i ) S ( N ) S S (0) , if S ( N − i +1) S ( N ) > S (0) S , S ( N − i ) S ( N ) ≤ S (0) S , , if S ( N − i ) S ( N ) > S (0) S . i = 2 , N , (274) a = ( − S ( N − S ( N ) , if S ( N − S ( N ) ≤ S (0) S , − S (0) S , if S ( N − S ( N ) > S (0) S (275) is true. The following equalities N Y i =1 (1 − a i ) = S (0) S g (cid:18) S (0) S ( N ) (cid:19) = S (0) S ( N ) , N − k Y i =1 (1 − a i ) = ( S ( k ) S ( N ) , if S ( k ) S ( N ) ≤ S (0) S , S (0) S , if S ( k ) S ( N ) > S (0) S , k = 1 , N − , (276) are valid. emark 3. Suppose that g ( x ) = x, x ∈ [0 , . Let us put g N − i ([ S ] N ) = g ( S ( i ) S ( N ) ) = S ( i ) S ( N ) , i = 0 , k, g N − i ([ S ] N ) = 1 , i = k + 1 , N − . Then, a = 1 − τ S (0) S , < τ ≤ , a i = 0 , i = 2 , N − k − ,a i = 1 − g i ([ S ] N ) g i − ([ S ] N ) , i = N − k, N , (277) is an estimation for the parameters a , . . . , a N . In the next Theorems we put τ = 1 . This corresponds to the fact that fair priceof super-hedge is minimal for the considered statistic.
Theorem 19.
On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177), with parameters a i , i = 1 , N , given by the formula(268). For the payoff function f ( x ) = ( x − K ) + , x ∈ (0 , ∞ ) , K > , the fair priceof super-hedge is given by the formula sup Q ∈ M E Q f ( S N ) = ( ( S − K ) + , if S (0) g N ([ S ] N ) ≥ K,S (cid:16) − S (0) g N ([ S ] N ) S (cid:17) , if S (0) g N ([ S ] N ) < K. (278) If S (0) g N ([ S ] N ) ≥ K, then the set of non arbitrage prices coincides with the point ( S − K ) + , in case if S (0) g N ([ S ] N ) < K the set of non arbitrage prices coincideswith the closed set h ( S − K ) + , S (cid:16) − S (0) g N ([ S ] N ) S (cid:17)i . The fair price of super-hedge for the statistic (274), (275) is given by the formula sup Q ∈ M E Q f ( S N ) = ( S − K ) + , if S S (0) S ( N ) ≥ K,S (cid:16) − S (0) S ( N ) (cid:17) , if S S (0) S ( N ) < K. (279) If S S (0) S ( N ) ≥ K, then the set of non arbitrage prices coincides with the point ( S − K ) + , in case if S S (0) S ( N ) < K the set of non arbitrage prices coincides with the closed set h ( S − K ) + , S (cid:16) − S (0) S ( N ) (cid:17)i . The fair price of super-hedge is minimal one for the statistic (268) with g i ( X N ) = g N ( X N ) = 1 , i = 1 , N − , and is given by the formula sup Q ∈ M E Q f ( S N ) = (cid:26) ( S − K ) + , if S (0) ≥ K,S − S (0) , if S (0) < K. (280) If S (0) ≥ K, then the set of non arbitrage prices coincides with the point ( S − K ) + , in case if S (0) < K the set of non arbitrage prices coincides with the closed set [( S − K ) + , S − S (0) ] . heorem 20. On the probability space { Ω N , F N , P N } , let the evolution of risky assetbe given by the formula (177) with the parameters a i , i = 1 , N , given by the formula(268). For the payoff function f ( x ) = ( K − x ) + , x ∈ (0 , ∞ ) , K > , the fair priceof super-hedge is given by the formula sup Q ∈ M E Q f ( S N ) = f (cid:0) S (0) g N ([ S ] N ) (cid:1) . (281) The set of non arbitrage prices coincides with the closed interval (cid:2) ( K − S ) + , f (cid:0) S (0) g N ([ S ] N ) (cid:1)(cid:3) . The fair price of super-hedge for the statistic (274), (275) is given by the formula sup Q ∈ M E Q f ( S N ) = f (cid:18) S S (0) S ( N ) (cid:19) . (282) The set of non arbitrage prices coincides with the closed interval h ( K − S ) + , f (cid:16) S S (0) S ( N ) (cid:17)i . The fair price of super-hedge is minimal one for the statistic (268) with g i ( X N ) = g N ( X N ) = 1 , i = 1 , N − , and is given by the formula sup Q ∈ M E Q f ( S N ) = f (cid:0) S (0) (cid:1) . (283) The set of non arbitrage prices coincides with the closed interval (cid:2) ( K − S ) + , f (cid:0) S (0) (cid:1)(cid:3) . Theorem 21.