Equivalent Locally Martingale Measure for the Deflator Process on Ordered Banach Algebra

)is paper aims at determining the measure ofQ under necessary and sufficient conditions.)emeasure is an equivalent measure for identifying the given P such that the process with respect to P is the deflator locally martingale. )e martingale and locally martingale measures will coincide for the deflator process discrete time. We define s-viable, s-price system, and no locally free lunch in ordered Banach algebra and identify that the s-price system (C, π) is s-viable if and only a character functional ψ|C ≤ π exists. We further demonstrate that no locally free lunch is a necessary and sufficient condition for the equivalent martingale measure Q to exist for the deflator process and the subcharacter φ ∈ Γ such that φ|C � π. )is paper proves the existence of more than one condition and that all conditions are equivalent.


Introduction
In this paper, we combine certain concepts in the functional analysis with other concepts in financial mathematics to generate new results. ese results are crucial to the improvement of efficient markets and the stock market. Czkwanianc and Pazkiewicz [1] highlighted the martingale measure for the stochastic process with discrete time; Harrison and Kreps [2] investigated the fundamental theorem and confirmed that the equivalent martingale measure is not sufficient to no-arbitrage alone for the stochastic process.
Many researchers have discussed the different methods for stopping time. Delbaen [3] introduced the martingale measure in continuous time and bounded. Kerps [4] presented a new definition of the no-arbitrage concept with certain properties. Dalang et al. [5] discussed the relationship between the equivalent martingale measure and no-arbitrage in stochastic securities. Harrison and Kreps [6] introduced the martingales and stochastic inferals in the theory of continues trading. Back and Pliska [7] identified the fundamental theorem of asset pricing with infinite state space. Schachermayer [8] employed the concept of no-arbitrage and built upon the work of David Kreps to identify the relation between equivalent measure and no-free lunch. Shachermayer [9] validated the fundamental theorem in Hilbert space. Delbaen and Shachermayer [10,11] developed the fundamental theorem of asset pricing for unbounded stochastic process.
Clark [12] created an equitable relationship between the martingale measure and the extension property while Schachermayer [13] determined the equivalences between the martingale measure in discrete time and arbitrage. Kabanov and Stricker [14] employed the equivalent martingale measure with bounded densities. Kabanov and Safarian [15] introduced the new properties for the function markets with transaction costs. Chen [16] introduced the viable costs and equilibrium price in frictional securities markets. Gaussel [17] identified the martingale property of prices when arbitrage opportunities can be found. Kabanov [18] built upon the studies of earlier researchers by introducing the concept of the locally martingale measure. Hussein [19,20] found the equivalent (super/sub) martingale measure and discussed the equivalent martingale in L P -space. Hussein and Fahim [21] proved new properties of the character in ordered Banach algebra. Prkaj and Ruf [22] determined the locally martingale measure in discrete time.
In this paper, we introduce the deflator process and determine the necessary and sufficient conditions of equivalent locally martingale for deflator process using ordered Banach algebra and other analytic concepts, such as algebra cone and character. We define certain concepts such as s-viable, s-price system, and s-no free lunch. We demonstrate the necessary and sufficient condition for the existence of the equivalent locally martingale measure. Different solutions have been introduced in relation to the topological conditions of arbitrage. Generally, some results validate the relation of no-arbitrage conditions to the existence of an equivalent locally martingale measure for deflator process. e triple (Ω, F, P) is called the probability space, where Ω is a nonempty, set, F is a σ-field on Ω, and P is a probability measure. e process S, sometimes denoted as (S t ) t∈I , is the process adapted to filter F t t∈I . e probability measure Q defined on F is equivalent to P if Q and P contain the same null sets. e equivalent probability measure Q is the equivalent martingale measure to (S t ) t∈I if (S t ) t∈I is martingale with respect to Q. us, if S is integral with respect to Q and for all t ∈ I, E((S t − S t− 1 )/F) � 0. A probability measure Q is an equivalent local martingale measure for S if Q is equivalent to P and S is Q-local martingale.
is paper is divided into four sections. Section 1 introduces the research. Section 2 explains the concepts and definitions required in the main subject as locally martingale, ordered Banach algebra, trading strategy, deflator process, subcharacter, s-price system, s-viable, and, s-no arbitrage. e classes of martingale and locally martingale measures coinciding with the discrete-time filtrated probability measure are discussed, and the existence of a one-to-one correspondence between equivalent martingale measure Q for the deflator process and the subcharacter φ ∈ Γ such that φ| C � π and some properties are proposed. In Section 3, the relationship between the equivalent locally martingale measure and the s-no-arbitrage is determined and the s-price system (C, π) is confirmed to be no-arbitrage if and only if a deflator process exists. An equivalent locally martingale measure exists if and only if a uniformly integrable deflator process exists. We prove that if S is a R d+1 -valued semimartingale with nonnegative components defined on the filtered probability space (Ω, F, {F t } t∈I ,P), an equivalent local martingale measure for deflator process also exists. Section 4 shows the relation between the deflator process and s-no-arbitrage, and we prove the uniformly integrable deflator process is necessary and sufficient condition to existence the equivalent locally martingale measure . Section 5 discusses the important conclusions of the research.
Definition 1 (see [22]). Let (Ω, F, P) be a complete probability space and {F t : t ≥ o} be a right continuous filtration. A right continuous adapted process {S t , {F t : t ≥ o} is locally martingale if there exists a sequence {τ n : n ≥ 1} of stopping time of filtration satisfying the following: For all α ≥ 0, w: lim n τ n � ∞ � ∪ n≥1 lim n τ n ≥ α � ∪ n≥1 B n,α differs from Ω by a P-null set and thus B n,α ∈ F α for all n ≥ 1 and B n,α ⊂ B n+1,α ↑Ω (almost everywhere). Hence, Definition 2 (see [23]). An algebra A is a linear space whose vectors can be multiplied in such a way that We speak of a complex or a real algebra according as the scalars are complex or real numbers. A commutative algebra is an algebra where multiplication satisfies the following condition: An algebra with identity is an algebra with the following property: there exists a nonzero element in the algebra, denoted by e and called the multiplicative identity element (or simply the identity), such that e · x � x · e � x, for all x.
Definition 3 (see [23]). If A is a normed linear space and also an algebra over C(or R), and if x · y ≤ x · y, for all x, y ∈ A, then A is called a complex (or real) normed algebra. If a normed algebra has the multiplicative identity, then we will postulate that ‖e‖ � 1.
A complex (real) Banach apace which is also a normed algebra is called Banach algebra.

Journal of Mathematics
Definition 4 (see [21]). Let A be a real Banach algebra with identity 1 and C nonempty subset of A. We call C a cone if the following hold: , then C will be called a proper cone. Any cone C on A induced an ordering ≥ on A as follows: We say that C is algebra cone if Definition 5 (Ordered Banach algebras, see [21]). A Banach algebra A with unite A is called ordered Banach algebra, which is denoted by (OBA) when A is ordered by a relation ≽ such that for every a, b ∈ A and β≽0, So, if A is ordered by an algebra cone C, we will obtain (A, C) as an ordered Banach algebra.
If A is an (OBA) and C is an algebra cone, C is called Theorem 1 (see [24]). Let M and N locally martingales and S and T be stopping times. en, (1)

Existence of the Equivalent Martingale Measure for the Deflator Process in Ordered Banach Algebra
We start the section by definition of trading strategy as follows: e process V t (H) is called value process of H.

Definition 8.
A functional π on ordered Banach algebra is a called a subcharacter if π is a sublinear functional on C and π(x · y) � π(x) · π(y). e set of all subcharacters is denoted by Γ.

Definition 9.
e s-price system is a pair (C, π) where C is an algebra cone of A and π is a subcharacter.

Definition 10.
e s-price system (C, π) is s-viable if π(x) ≤ 0 and x ± y for all y ∈ C such that π(y) ≤ 0, in a special case if e ≥ x with π(x) ≤ 0.

Theorem 2.
e classes of the martingale and the locally martingale coincide on the discrete-time filtrated probability measure.
Proof. Take M � (M i ) i≤n as a A n -valued local martingale and H � (H i ) i≤n as a A n -valued predictable process. We define If H i is bounded and M i is martingale, then S is martingale.
Otherwise, let {τ n } be a sequence of stopping times increasing as to ∞, such that each process M τ n is a martingale and the process S τ n is martingale.
Let S be martingale; that is, S τ n n with the sequence τ n each is integrable and { } as F n -measurable integrable and [u n | F n ] � 0. en, M n � n i�1 u i is a martingale, and (4) holds.

□ Theorem 3. Let A be ordered Banach algebra with algebra cone (A, C). e s-price system (C, π) is s-viable if and only if a character functional ψ| C ≤ π exists.
Proof. Suppose there exists a character functional ψ| C ≤ π; Conversely, suppose that s-price system is s-viable.
If h ∈ M ∩ N, then h > e and π(h) ≤ 0; this contradicts that (C, π) is s-viable. en, M and N are disjoint.
For δ ∈ R, m α ∈ C. Proposition 1. e following conditions are equivalent: (1) e s-price system (C, π) is no locally free lunch (2) C ∩ X + ≠ ϕ, where C is the set of all limit points of convergence net in C Proof. Suppose that (C, π) admits an s-locally free lunch. en, a net (m α , x α ): α ∈ Λ ⊆ C × X and x ∈ X + exists such that α ∈ Λ and for δ ∈ R en, m α ∈ C because m α ≥ x α , and x α − m α ∈ C for all α ∈ Λ.
x α ⟶ τ x and m α ⟶ m, and then, x − m ∈ C; we have x ≽ m, that is, m ∈ C.
x ∈ X + and x ≽ e and x ≽ m; we obtain m ≽ e, which implies that m ∈ m ∈ C ∧ m ∈ X + .

Theorem 4. If no locally free lunch exists, then a one-to-one correspondence between the equivalent martingale measure Q for the deflator process and the subcharacter φ ∈ Γ exists such that φ| C � π. is correspondence is given by Q (A) � φ(I A ) and φ(x) � E Q (x).
Proof. Let Q be an equivalent measure; set ρ � (dQ/dP).
for all x ∈ X. en, ψ(x) � E(ρx) and ρ ∈ L 2 ; ψ is linear functional and continuous: en, ψ is character. Q ∼ P and ρ are strictly positive, and thus, ψ is strictly positive.
φ is continuous, so Q is a σ-additive measure, and dQ/ dP � ρ is square-integrable.
us, Q is probability measure equivalent to P.

Relation between Equivalent Locally Martingale Measure and S-No-Arbitrage
In this section, we introduce important results related to the main subject.
Conversely, suppose a deflator process exists. is is equivalent to S n ∈ K, that is, S 0 ≻0. X 0 � 1 and Proof. A measure Q equivalent to P with Radon-Nikodom derivative dQ t /dP t � Z t is strictly positive and E[Z t ] � 1, for a given sequence τ n with finite stopping time where E(Sχ τ n >v and ZS is itself a local martingale. en, for any S ∈ M, π(S) � E 0 P (S/X) � E Q (S) and en, S i is uniformly integrable. Conversely, if {S i } is a uniformly integrable deflator process. en, S i ⟶ Z t ∈ ℓ 1 (P) and E[Z t ] � 1.
Choose Q as a probability measure on the space (Ω, F)equivalent to P; then, with Radon-Nikodym theorem, it becomes (dQ t /dP t ) � Z t , so that Q u : For any S ∈ M, E 0 P (S/X) � π(S), but π(S) � E Q (S). en, E 0 P (S/X) � E Q (S) and the P-martingale property of S is Q-martingale property of S. □ Theorem 6. Let S be a R d+1 -valued semimartingale with nonnegative components defined on the filtered probability space (Ω, F, {F t } t∈I , P). en, equivalent local martingale measure for deflator process exists.
Proof. Let Θ be a set of all nonnegative processes, D � X∈ Θ: X T > 0 and ≼ be a binary relation on Θ defined as is order is not preferences ordered, if we have A to the quotient space obtained from Θ by identifying processes whose difference is locally martingale null at 0; the pair (A, ≼) is an ordered linear space. Let C be a set of all X in A such that X ≤ 1.
en, the closure C � X∈ Θ: E [XY]≼1 for all Y ∈ D}. Define T: C ⟶ R as follows: en, C ∩ A ++ � ∅, because for every X∈ C, we have q ≤ 0 for some q ∈P(X); let X°∈ A ++ . en, X ∉ C. But C − A + ⊆ C implies C − A + ⊆C. us, X ∉ C − A + . So there exists an open, convex neighborhood of X. Say N(X), such that N(X) ∩ (C − A + ) � ∅.
In particular, Let C(X) denote the convex cone generated by (N(X)) by C(X) � {Y∈ A: Y � λX for some λ > 0 and X∈ N(X)}.
en, C(X 0 ) ∩ (C − A + ) � ∅, and by Hahn-Banach theorem, a continuous linear functional P: A ⟶ R such that P(X) ≥ 0 for every X∈ C(X) and P(X) ≤ 0 for every X∈ C − A + .In particular, P(X) ≤ 0 for every X∈ C. Hence, we obtain P: C ⟶ R as a strictly positive character.
For every X∈ C, P(X) ≤ 0 implies P(X) ≤ 0 replacing m with − X in this implication.
We see that for every X∈ C P(X) ≥ 0 and P(X) ≥ 0, P(X) � 0 implies P(X) � 0 for every X∈ C, and there exists some λ ∈ R such that P(X) � λP(X) for every X∈ C, X 0 ∈ C ++ , and P(X 0 ) > 0; P is continuous on C. Hence, P: C ⟶ R is continuous, positive linear functional.
Journal of Mathematics □

Conclusion
In this paper, we use ordered Banach algebra and the algebra cone to define the deflator random variable and confirm that the s-price system (C, π) is no locally free lunch and equivalent to C ∩ X + ≠ ϕ, where C is the set of all limit points of convergence net in C and the s-price system (C, π) is s-viable if and only if a subcharacter functional ψ| C ≤ π exists. We validate that the s-price system (C, π) is no-s-arbitrage if and only if a deflator process exists and confirm that the trading strategy H is self-financing with respect to the deflator process if and only if H is self-financing with respect to S(t). No locally free lunch is a necessary condition to materialize a one to-one correspondence between the equivalent martingale measure Q for the deflator process and the subcharacter φ ∈ Γ such that φ| C � π. e uniformly integrable deflator process is also an important condition for generating an equivalent locally martingale measure and to prove that if X is a R d+1 -valued semi martingale with nonnegative components defined on the filtered probability space (Ω, F, {F t } t∈I , and P), we can construct an equivalent local martingale measure for the deflator process.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.