Abstract

This paper aims at determining the measure of Q under necessary and sufficient conditions. The measure is an equivalent measure for identifying the given P such that the process with respect to P is the deflator locally martingale. The 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 is s-viable if and only a character functional 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 such that . This paper proves the existence of more than one condition and that all conditions are equivalent.

1. Introduction

In this paper, we combine certain concepts in the functional analysis with other concepts in financial mathematics to generate new results. These 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 -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.

The 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. The process S, sometimes denoted as , is the process adapted to filter . The probability measure Q defined on F is equivalent to P if Q and P contain the same null sets. The equivalent probability measure Q is the equivalent martingale measure to if is martingale with respect to Q. Thus, if S is integral with respect to Q and for all t ∈ I, . A probability measure Q is an equivalent local martingale measure for S if Q is equivalent to P and S is Q-local martingale.

This 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. The 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 such that 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 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.

2. Preliminary

Consider a filtered probability space (Ω, F,{F_t}_t∈I, P) where with 0, T ∈ I, for a fixed T < ∞. Let F₀ = {F, Ω}, F_T = F, S = (S(t))_t∈I be a ℜ^d+1-valued stochastic process with components S_i(t) for i = 0, 1, …, d and satisfy the following properties:(1)S_iis an adapted to {F_t}_t∈I for all i = 0, 1, …, d(2)E((S_i)²(t)) < ∞ for t ∈ I and for all i = 0, 1, …, d(3)S₀(t, ω) = 1 for all t ∈ I and for all ω ∈ Ω.

Definition 1. (see [22]). Let (Ω, F, P) be a complete probability space and {F_t: } be a right continuous filtration. A right continuous adapted process {S_t, {F_t: } is locally martingale if there exists a sequence {: n ≥ 1} of stopping time of filtration satisfying the following:(1)(2)(3)(4)If Then, for each n, {, F_t, } is a uniformly integrable martingale. This concept plays a key role in the analysis. A positive local martingale {, F_t, } is super martingale, that is, let , . Now, the set .
For all ,  =  differs from by a P-null set and thus for all and (almost everywhere).
Hence,because  =  on

Definition 2. (see [23]). An algebra is a linear space whose vectors can be multiplied in such a way that(1)(2) and (3) for all scalar 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 and called the multiplicative identity element (or simply the identity), such that .

Definition 3. (see [23]). If is a normed linear space and also an algebra over , and if then is called a complex (or real) normed algebra. If a normed algebra has the multiplicative identity, then we will postulate that .
A complex (real) Banach apace which is also a normed algebra is called Banach algebra.

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:(1)a + b  C for all a, b  C(2) for all a  C and If  = {0}, then C will be called a proper cone. Any cone C on A induced an ordering ≥ on as follows: if and only if for all .
We say that C is algebra cone if(1) for all (2)

Definition 5. (Ordered Banach algebras, see [21]). 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 and ,(1)(2)(3)(4)So, if is ordered by an algebra cone C, we will obtain (, C) as an ordered Banach algebra.
If is an (OBA) and C is an algebra cone, C is called normal if there is , for any .

Theorem 1. (see [24]). Let M and N locally martingales and S and T be stopping times. Then,(1)The sum M + N is also local martingale(2)If T reduces M and , then S reduces M(3)If both S and T reduce, then also reduces(4)The processes M^T and are local martingale

3. 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:

Definition 6. A trading strategy is an -valued , such that is a integrable with respect to semimartingale .
Let , where .
The process is called value process of .

Definition 7. Let , be a strictly positive semimartingale; we set and as a sequence of random variables, which is strictly positive for . We call a deflator random variable, and a measurable function is called deflator process.

Definition 8. A functional on ordered Banach algebra is a called a subcharacter if is a sublinear functional on and . The set of all subcharacters is denoted by Γ.

Definition 9. The s-price system is a pair where C is an algebra cone of and .

Definition 10. The s-price system is s-viable if in a special case if

Definition 11. Let be (OBA) with the algebra cone . Topology is weak topology, is a strictly positive linear functional, and is s-no-arbitrage; if , then .

Theorem 2. The classes of the martingale and the locally martingale coincide on the discrete-time filtrated probability measure.

Proof. Take M =  as a -valued local martingale and as a -valued predictable process. We defineIf is bounded and is martingale, then S is martingale.
Otherwise, let {} be a sequence of stopping times increasing as to , such that each process is a martingale and the process is martingale.
Let be martingale; that is, with the sequence each is integrable and on the -measurable set {}.
. Take as -measurable integrable and . Then, is a martingale, and (4) holds.

Theorem 3. Let A be ordered Banach algebra with algebra cone (A, C). The s-price system is s-viable if and only if a character functional exists.

Proof. Suppose there exists a character functional ; let Because then . That is, . Hence, ; that is, is s-viable.
Conversely, suppose that s-price system is s-viable.
Define M = . If , then ; this contradicts that is s-viable. Then,
Because is cone, then C is a convex set. To prove N is cone, let h₁, h₂ ∈ C and 0 ≤ λ, 0 ≤ β, π(h₁) ≤ 0, and π(h₂) ≤ 0; that is, λπ(h₁) ≤ 0 and π(h₂) ≤ 0. Then, λπ(h₁) + ()π(h₂) ≤ 0; that is, π(λh₁ + h₂) ≤ 0. π is character functional; then λh₁ + h₂ ∈ C and is cone.
Thus, are disjointed, nonempty convex sets in (OBA). Using the separation theorem, there exists a nontrivial continuous linear functional ψ on such that ψ(h) > 0 for all h ∈ and ψ(h) ≤ 0 for all .
If , then or .
By taking h ∈ M and  ∈ N, then ψ(h). ψ() ≤ 0; that is, , to prove ψ(h₀) > 0. ψ is nontrivial, and thus, x₀ ∈  exists, such that ψ(h₀) > 0.
h₀ − λx₀ ≻  for and h₀ ∈ H. Therefore, , and by linearity of ψ, we have , and when , we get . As and , ψ can be normalized so that ψ(h₀) = π(h₀).
To prove , let h ∈ M. If and such that ,    +  ≤  for h, h₀ ∈ C, and C is cone, then we have . Therefore, , ; that is, ψ(h)  λψ(h₀) = −λπ(h₀) = π(h).

Definition 12. An s-price is said to be s-no locally free lunch if a net and exist, such that(1).(2), and such that . For , m_α ∈ C.

Proposition 1. The following conditions are equivalent:(1)The s-price system (C, π) is no locally free lunch(2), where is the set of all limit points of convergence net in C

Proof. Suppose that (C, π) admits an s-locally free lunch. Then, a net and x ∈ X₊ exists such that(1)(2) such that for all α ∈ Λ and for Then, m_α ∈ C because , and for all α ∈ Λ.
and then, x − m ∈ C; we have x ≽ m, that is, m ∈ .
x ∈ X₊ and x ≽ e and x ≽ ; we obtain m ≽ e, which implies that m ∈ .
Hence, m ∈ ; that is, .
Conversely, suppose . Then, y ∈  exists, that is, y ∈  and y ∈ X₊.
x _α ∈ C such that and m_α ∈ C such that x_α ≥ m_α for all α ∈ Λ. m_α ∈ C and , for all α ∈ Λ.

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  ∈ Γ exists such that . This correspondence is given by Q (A) = (I_A) and (x) = E_Q(x).

Proof. Let Q be an equivalent measure; set
Define . Then, and ; is linear functional and continuous:Then, is character. Q ∼ P and ρ are strictly positive, and thus, is strictly positive.
Take  = ψ, that is,  ∈ Γ; this remains to show that .
For n = 1, 2, …, k, we have :Because is a martingale with respect to Q, this equality yields . ; that is, ; then, , π(h) =  = , and thus E_Q(h) = π(h), ψ(h) = π(h) for all m ∈ C.
Conversely, let  ∈ Γ such that . Define Q: F ⟶ ℜ by Q(X) = (I_X) for all X ∈ F.
If P(X) = 0, that is, I_X = 0, then Q(X) =  = 0.
If P(X) > 0, then I_X ∈ H₊. Therefore, ϕ(I_X) > 0, that is, Q(X) > 0.
To prove that Q is measure Q() = (I_∅) = 0 and , where  ∈ F =  =  ≤ , then Q is measure. Also, to prove the measure Q is equivalent to measure P.
.
is continuous, so Q is a σ-additive measure, and dQ/dP = ρ is square-integrable.
I _Ω = 1 if x ∈  and 0 if x ∉ ; then, (I_x) = 1; that is, Q() = 1. Thus, Q is probability measure equivalent to P.

Proposition 2. A trading strategy is self-financing with respect to the deflator process if and only if is self-financing with respect to .

Proof. For any , for all a sequence of strictly positive random variables exists as follows:Then, is self-financing with respect to the deflator process .

4. Relation between Equivalent Locally Martingale Measure and S-No-Arbitrage

In this section, we introduce important results related to the main subject.

Proposition 3. Let (Ω, F, P) be a probability space. Then, is s-no-arbitrage if and only if a deflator process exists.

Proof. Suppose is no-arbitrage. If , that is, .
Also, . For any process ; .
Conversely, suppose a deflator process exists. This is equivalent to , that is, . and . Then, .