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

Authors: Boushra Y. Hussein
Journal: Journal of Mathematics
 (2020)

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## Abstract

This paper aims at determining the measure ofQ 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 C,π is s-viable if and only a character functional ψC≤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 φC=π. This paper proves the existence of more than one condition and that all conditions are equivalent.


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## Body

## 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 LP-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 ass-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 Stt∈I, is the process adapted to filter Ftt∈I. 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 toStt∈I if Stt∈I is martingale with respect to Q. Thus, if S is integral with respect to Q and for all t ∈ I, ESt−St−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.This paper is divided into four sections. Section1 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 φ∈Γ such that φC=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 ℜd+1-valued semimartingale with nonnegative components defined on the filtered probability space (Ω, F, {Ft}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,{Ft}t∈I, P) where I⊂0,1 with 0, T ∈ I, for a fixed T < ∞. Let F0 = {F, Ω}, FT = F, S = (S(t))t∈I be a ℜd+1-valued stochastic process with components Si(t) for i = 0, 1, …, d and satisfy the following properties:(1)
Siis an adapted to {Ft}t∈I for all i = 0, 1, …, d(2)
E((Si)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 {Ft: t≥o} be a right continuous filtration. A right continuous adapted process {St, {Ft: t≥o} is locally martingale if there exists a sequence {τn: n ≥ 1} of stopping time of filtration satisfying the following:(1)
τn≤τn+1(2)
Pτn≤n=1(3)
Plimnτn=∞=1(4)
Ifτtn=Tn∧t,Stn=S∘τtn
Then, for eachn, {Stn, Ft, t≥0} is a uniformly integrable martingale. This concept plays a key role in the analysis. A positive local martingale {Stn, Ft, t≥0} is super martingale, that is, let A∈Fs, 0≤s≤t. Now, the set X∘τtn,t≥0is uniformly integrable.
For allα≥0, w:limnτn=∞ = ∪n≥1limnτn≥α=∪n≥1Bn,α differs from Ω by a P-null set and thus Bn,α∈Fα for all n≥1 and Bn,α⊂Bn+1,α↑Ω (almost everywhere).
Hence,(1)∫AXtdP=limn∫A∩Bn,tXtdP=limn∫A∩Bn,tXt°τtndP,because X∘τtn = Xt on(2)Bn,t≤limn∫A∩Bn,sXt°τtndPBn,t⊂Bn,sfors<tandXt≥0=limn∫A∩Bn,sXt°τndP,by martingale property ofX∘τsn,s≥0=∫A∩Bn,tXtdP=∫AXtdP,thenEXtFs≤Xs.Definition 2.
(see [23]). An algebra A is a linear space whose vectors can be multiplied in such a way that(1)
x⋅y⋅z=x⋅y⋅z(2)
x+yz=xz+yz and zx+y=zx+zy(3)
αxy=αxy=xαy for all scalar α
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:(3)xy=yx,for allx,y∈A.
An algebra with identity is an algebra with the following property: there exists a nonzero element in the algebra, denoted bye and called the multiplicative identity element (or simply the identity), such that e⋅x=x⋅e=x,for allx.Definition 3.
(see [23]). If A is a normed linear space and also an algebra over ℂorℝ, and if x⋅y≤x⋅y,for allx,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.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)
λa∈C for all a ∈ C and λ≥0
IfC∩−C = {0}, then C will be called a proper cone. Any cone C on A induced an ordering ≥ on A as follows: b≽a if and only if b−a∈C for all a,b∈C.
We say thatC is algebra cone if(1)
a⋅b∈C for all a,b∈C(2)
e∈CDefinition 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,(1)
a,b≽0⇒a+b≽0(2)
a≽0,β≽0⇒λa≽0(3)
a,b≽0⇒ab≽0(4)
e≽0
So, ifA is ordered by an algebra cone C, we will obtain (A, C) as an ordered Banach algebra.
IfA is an (OBA) and C is an algebra cone, C is called normal if there is β>0, for any a,b∈A,0≤a≤b<⇒a≤βb.Theorem 1.
(see [24]). Let M and N locally martingales and S and T be stopping times. Then,(1)
The sumM + N is also local martingale(2)
IfT reduces M and S⊆T, then S reduces M(3)
If bothS and T reduce, then S∪T also reduces(4)
The processesMT and MT>0T 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 strategyH is an Rd+1-valued vector stochastic process,H=φt0,…,φtd+1t=1T, such that φt is a Ft−1−measurable function for allt≥1, integrable with respect to semimartingale M=Stt=1T.
LetVtH=H⋅M=φt0St0+φt,St=φt0St0+∑j=1mφtjStj, where H=φt0,…,φtd+1.
The processVtH is called value process of H.Definition 7.
LetSi,i=0,1,…,m, be a strictly positive semimartingale; we set S=S1,S2,…,Sm and Xtt=0T as a sequence of random variables, which is strictly positive for t∈0,1,…,T. We call Xtt=0T−1 a deflator random variable, and a measurable function S˜=S1/Xt,S2/Xt,…,Sm/Xt is called deflator process.Definition 8.
A functionalπ on ordered Banach algebra is a called a subcharacter if π is a sublinear functional on C and πx⋅y=πx⋅πy. The set of all subcharacters is denoted by Γ.Definition 9.
The s-price system is a pairC,π where C is an algebra cone of A and πis a subcharacter.Definition 10.
The s-price systemC,π is s-viable if πx≤0andx±yfor ally∈Csuch thatπy≤0, in a special case if e≥xwithπx≤0.Definition 11.
LetA,C be (OBA) with the algebra cone C=A+∖0,K0⊆X,andK=spanK0. Topology τ is weak topology, π:C⟶ℝ is a strictly positive linear functional, and C,π is s-no-arbitrage; if x∈K∩C, then πx>0.Theorem 2.
The classes of the martingale and the locally martingale coincide on the discrete-time filtrated probability measure.Proof.
TakeM = Mii≤n as a An-valued local martingale and H=Hii≤n as a An-valued predictable process. We define(4)S=H⋅M=φt,St=∑i=1nHiMi.
IfHi is bounded and Mi is martingale, then S is martingale.
Otherwise, let {τn} be a sequence of stopping times increasing as to ∞, such that each process Mτn is a martingale and the process Sτn is martingale.
LetS be martingale; that is, Snτn with the sequence τn each is integrable and ESn+1Fn=ESn+1τnFn on the Fn-measurable set {τn>n}.
ESn+1Fn=ESn+1τnFn=Snτn=Sn. Take un=∑k≥01/k+1ΔSnIw:EΔSnFn−1w∈k,k+1 as Fn-measurable integrable and unFn=0. Then, Mn=∑i=1nui is a martingale, and (4) holds.Theorem 3.
LetA be ordered Banach algebra with algebra cone (A, C). The s-price system C,π is s-viable if and only if a character functional ψC≤π exists.Proof.
Suppose there exists a character functionalψC≤π; let h∈Cwithπh≤0. Because ψh≤πhfor allh∈H, then ψh≤0=ψ0. That is, ψ0≥ψh. Hence, h≥e; that is, C,π is s-viable.
Conversely, suppose that s-price system is s-viable.
DefineM = h∈A:h>e,N=h∈C:πh≤0. If h∈M∩N, then h>eandπh≤0; this contradicts that C,π is s-viable. Then, MandNare disjoint.
BecauseC 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 βπ(h2) ≤ 0. Then, λπ(h1) + (β)π(h2) ≤ 0; that is, π(λh1 + βh2) ≤ 0. π is character functional; then λh1 + βh2 ∈ C and N is cone.
Thus,MandN are disjointed, nonempty convex sets in (OBA). Using the separation theorem, there exists a nontrivial continuous linear functional ψ on A such that ψ(h) > 0 for all h ∈ Mand ψ(h) ≤ 0 for all h∈N.
Ifψh⋅g≠ψh⋅ψg, then ψh⋅g>ψh⋅ψg or ψh⋅g<ψh⋅ψg.
By takingh ∈ M and g ∈ N, then ψ(h). ψ(g) ≤ 0; that is, ψh⋅g≤0, to prove ψ(h0) > 0. ψ is nontrivial, and thus, x0 ∈ A exists, such that ψ(h0) > 0.
Mis cone and we haveh0 − λx0 ≻ θ for λ>o and h0 ∈ H. Therefore, ψh0−λx0≥0, and by linearity of ψ, we have ψm0−λψx0≥0, and when ψm0≥λψx0>0, we get ψh0>0. As πh0>0 and ψh0>0, ψ can be normalized so that ψ(h0) = π(h0).
To proveψM≤π, let h ∈ M. If πh≤0 and λ>0 such that πh+λπh0=0, π1/λ+1+λ/λ+1h0 ≤ 1/λ+1πh + λ/λ+1πh0 ≤ 1/λ+1πh+λπh0=0 for h, h0 ∈ C, and C is cone, then we have 1/λ+1h+λ/λ+1h0∈C. Therefore, ψ1/λ+1h+λ/λ+1h0≤0, 1/λ+1ψh+λ/λ+1ψh0≤0; 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 netmα,xα:α∈Λ⊆C×A and x∈X+ exist, such that(1)
mα≥xα∀α∈Λ.(2)
xα⟶τx, and mα⟶m such that πm≤δ∀α∈Λ. For δ∈R, mα ∈ C.Proposition 1.
The following conditions are equivalent:(1)
The 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 CProof.
Suppose that(C, π) admits an s-locally free lunch. Then, a net mα,xα:α∈Λ⊆C×X and x ∈ X+ exists such that(1)
mα≥xα∀α∈Λ(2)
xα⟶τxandmα⟶m such that πm≤δ for all α ∈ Λ and for δ∈ℝ
Then,mα ∈ C because mα≥xα, and xα−mα∈C for all α ∈ Λ.
xα⟶τxandmα⟶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+.
Hence,m ∈ C˜∩X+; that is, C˜∩X+≠ϕ.
Conversely, supposeC˜∩X+≠ϕ. Then, y ∈ C˜∩X+ exists, that is, y ∈ C˜ and y ∈ X+.
xα ∈ C such that xα⟶τy and mα ∈ C such that xα ≥ mα for all α ∈ Λ. mα ∈ C and πm≤δ, for all α ∈ Λ.Theorem 4.
If no locally free lunch exists, then a one-to- one correspondence between the equivalent martingale measureQ for the deflator process and the subcharacter φ ∈ Γ exists such that φC=π. This correspondence is given by Q (A) = φ(IA) and φ(x) = EQ(x).Proof.
LetQ be an equivalent measure; set ρ=dQ/dP.
Defineψ:X⟶Rbyψx=EQxfor allx∈X. Then, ψx=Eρx and ρ∈L2; ψ is linear functional and continuous:(5)ψx⋅y=EQx⋅y=EQx⋅EQy=ψx⋅ψy.
Then,ψ is character. Q ∼ P and ρ are strictly positive, and thus, ψ is strictly positive.
Takeφ = ψ, that is, φ ∈ Γ; this remains to show that ψM=π.
Forn = 1, 2, …, k, we have EQϑtn,S˜tn/Ftn−1=EQϑtn−1,S˜tn/Ftn−1:(6)ϑtn,S˜tn=ϑtn−1,S˜tn,EQϑtn,S˜tnFtn−1=ϑtn−1,EQS˜tnFtn−1=ϑtn−1,S˜tn−1.
BecauseS˜ is a martingale with respect to Q, this equality yields EQϑT,S˜T=ϑ0,S˜0. VϑT=h; that is, ϑT,S˜T=h; then, EQh=ϑ0,S˜0, π(h) = Vϑ0 = ϑ0,S˜0, and thus EQ(h) = π(h), ψ(h) = π(h) for all m ∈ C.
Conversely, letφ ∈ Γ such that φM=π. Define Q: F ⟶ ℜ by Q(X) = φ(IX) for all X ∈ F.
IfP(X) = 0, that is, IX = 0, then Q(X) = φ0 = 0.
IfP(X) > 0, then IX ∈ H+. Therefore, ϕ(IX) > 0, that is, Q(X) > 0.
To prove thatQ is measure Q(ϕ) = φ(I∅) = 0 and Q∪i=1∞Ei, where Ei ∈ F = φI∪i=1∞Ei=φ∑i=1∞IEi−∑i≠j=1∞IEi∩Ej = φλ∑i=1∞IEi+1−λ∑i≠=1∞IEjEi∩Ej=φϕ ≤ ∑i=1∞φIEi=∑i=1∞QIEi, then Q is measure. Also, to prove the measure Q is equivalent to measure P.
IfPX>0,thenIXis positive linear functional,φIX>0;that is,QX>0.
φ is continuous, so Q is a σ-additive measure, and dQ/dP = ρ is square-integrable.
IΩ = 1 if x ∈ Ω and 0 if x ∉ Ω; then, φ(Ix) = 1; that is, Q(Ω) = 1. Thus, Q is probability measure equivalent to P.Proposition 2.
A trading strategyH is self-financing with respect to the deflator process if and only if H is self-financing with respect to St.Proof.
(7)VtH=Vt+1H,φt0St0+φt,St=φt+10St+10+φt+1,St+1,φt0St0+∑j=1mφtjStj=φt+10St+10+∑j=1mφt+1jSt+1j.
For anyXtt=0T, for all t∈0,1,…,T, a sequence of strictly positive random variables exists as follows:(8)φt0Xt−1St0+∑j=1mφtjXt−1Stj=φt+10Xt−1St+10+∑j=1mφt+1jXt−1St+1jφt0S˜t0+∑j=1mφtjS˜tj=φt+10S˜t+10+∑j=1mφt+1jS˜t+10.
Then,H is self-financing with respect to the deflator process S˜t.


## 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, C,π is s-no-arbitrage if and only if a deflator process exists.Proof.
SupposeC,π is no-arbitrage. If S˜∈K∩C, that is, S˜≻0.
Also,S0=1. For any process X∈M,the deflator processS˜t=St/Xt; then,πS˜=ES˜n.
Conversely, suppose a deflator process exists. This is equivalent toSn∈K, that is, S0≻0. X0=1 and S˜0=S0/X0=ESn/Xn=ES˜n. Then, πS˜n=S˜0≻0.Theorem 5.
Let (Ω,F, P) be a probability space. Then, an equivalent locally martingale exists if and only if a uniformly integrable deflator process exists.Proof.
A measureQ equivalent to P with Radon–Nikodom derivative dQt/dPt = Zt is strictly positive and EZt=1, for a given sequenceτn with finite stopping time vF∈Fv, F∩τn>v∈Fv∩Fτn=Fv⋏τn:(9)EχFES˜χτn>vFv∧τn=ES˜Fv∧τnχF∩τn>v=EES˜χF∩τn>vFv∧τnES˜χF∩τn>v,where ES˜χτn>vFv∧τn is Fv−measurable,ES˜χτn>vFv∧τn=ES˜χτn>vFv, that is, ES˜Fv∧τnχτn>v=ES˜Fvχτn>v, and ZS˜ is itself a local martingale.
Then, for anyS∈M, πS˜=EP0S/X=EQS˜ and S˜i = EPiZt, lim E[ZtS˜i] = E[Zt] = 1.
Then,S˜i is uniformly integrable.
Conversely, if {S˜i} is a uniformly integrable deflator process. Then, S˜i⟶Zt∈ℓ1P and E[Zt] = 1.
ChooseQ as a probability measure on the space Ω,Fequivalent toP;then, with Radon–Nikodym theorem, it becomes dQt/dPt=Zt, so that Qu:=Qu=ZtuPu for u∈A.
For anyS∈M, EP0S/X=πS˜, but πS˜=EQS˜. Then, EP0S/X=EQS˜ and the P-martingale property of S˜ is Q-martingale property of S.Theorem 6.
LetS be a ℜd+1-valued semimartingale with nonnegative components defined on the filtered probability space (Ω, F, {Ft}t∈I, P). Then, equivalent local martingale measure for deflator process exists.Proof.
LetΘ be a set of all nonnegative processes, D = X˜∈Θ:X˜T>0 and ≼ be a binary relation on Θ defined as(10)X˜≼Y˜iffX˜0≤Y˜0andX˜−Y˜is a supermartingale.
This order is not preferences ordered, if we haveA 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˜inAsuch thatX˜≤1. Then, the closure C¯ = X˜∈Θ:EX˜Y˜≼1for allY˜∈D. Define T: C¯⟶R as follows: T(X˜) = EX˜for allX˜∈C¯.
Then,C¯∩A++=∅, because for every X˜∈C¯, we have q≤0 for some q∈P¯X˜; let X˜°∈A++. Then, X˜∉C¯. But C−A+⊆C implies C−A+¯⊆C¯.Thus, X˜∉C−A+¯. So there exists an open, convex neighborhood of X˜. Say NX˜, such that NX˜∩C−A+¯=∅. In particular, NX˜∩C−A+=∅.
LetCX˜ denote the convex cone generated by NX˜ by CX˜ = {Y˜∈Α:Y˜=λX˜ for some λ>0 and X˜∈NX˜}. Then, CX˜0∩C−A+=∅, and by Hahn–Banach theorem, a continuous linear functional P:A⟶R such that PX˜≥0 for every X˜∈CX˜ and PX˜≤0 for every X˜∈C−A+.In particular, PX˜≤0 for every X˜∈C¯. Hence, we obtain P¯:C¯⟶R as a strictly positive character.
For everyX˜∈C, PX˜≤0 implies PX˜≤0 replacing m with −X˜ in this implication.
We see that for everyX˜∈CPX˜≥0 and PX˜≥0, PX˜=0 implies PX˜=0 for every X˜∈C, and there exists some λ∈R such that PX˜=λPX˜ for every X˜∈C, X˜0∈C++, and PX˜0>0; P¯ is continuous on C. Hence, P¯:C¯⟶R is continuous, positive linear functional.
Finally, we show thatP¯ is strictly positive. First, assume that P¯ is not strictly positive. Then, there exists X˜∈C¯++ such that P¯X˜=0. Let X˜α:α∈Γ be a net in C such that X˜α⟶X˜. P¯X˜α⟶P¯X˜=0. So, PX˜α>0 for all α sufficiently large in the directed index set.
Defineβα=PX˜α/PX˜0 for every α∈Γ, βα>0 for all α sufficiently large, and βα⟶0.
DefineX˜α∗=X˜α−βαX˜0 for every y∈Γ and X˜α∗∈C.
Also,PX˜α∗=PX˜α−βαPX˜0=0 implies X˜0∈C¯, and thus X˜0∈C¯∩A¯++ is a contradiction. P¯ is strictly positive. To prove any process under condition is locally martingale, let MQ be a set of all measures Q equivalent to measure P, by taking YtQ=EdQ/dPFt. At is a nondecreasing, adapted process such that A0 = 0, and we assume X0=1 in [0, T] and find H to be a predictable S-integrable process such that X=1+∫0tHvdSv. We set Y˜v=Xs/Xs−AsHv, so that Y is an S-integrable predictable process:(11)1+∫0tHvdSv=Xv,v≤s,Xs+XsXs−AsXv−Xs,v>s.
Xs+Xs/Xs−AsXv−Av−Xs−As = XsXv−Av/Xs−Av≥0; thus, this process is supermartingale:(12)Y˜tQXs=Ys˜1+∫0tHvdSv≥EYs˜1+∫0tHvdSvFt=XsXs−AsEYs˜Xt−AsFs≥XsXs−AsEYs˜Xt−AtFt≥XsXs−AsEYs˜Xt−AtFS.
If we multiply byXs−As/Xs, we can obtain the desired supermartingale.


## 5. Conclusion

In this paper, we use ordered Banach algebra and the algebra cone to define the deflator random variable and confirm that thes-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 ψC≤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 St. 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 φC=π. The uniformly integrable deflator process is also an important condition for generating an equivalent locally martingale measure and to prove that if X is a ℜd+1-valued semi martingale with nonnegative components defined on the filtered probability space (Ω, F, {Ft}t∈I, and P), we can construct an equivalent local martingale measure for the deflator process.



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*Source: 5785098-2020-06-09.xml*