Strictly Positive Functionals in Orlicz Spaces

1. Introduction

The recent evolution of the Asset Pricing Theory is established in the following Conditions, which may be summarized in the following way: In [4] No -Market -Free Lunch condition about pricing functionals was used to prove equivalence between No-Free Lunch, and the existence of equivalent (local) martingale measures $\mathbb{Q}$. In [6], Orlicz spaces define the property of No Market Free Lunch in $L^{\infty }$, in order to deduce the equivalence between Kreps No-Free Lunch and the existence of equivalent (local) martingale measures $\mathbb{Q}$, relying on the topological results from [9]. We call Kreps No -Free Lunch the No-Free Luch condition as it appears in [3], due to the seminal work of [7]. Here, we examine some separate families of Young functions in order to produce similar theorems, which are valid for both of dual pairs $〈M^{\Phi },L^{\Psi }〉$ and $〈L^{\Phi },L^{\Phi }〉$ of Orlicz spaces.

We denote by $E^{*}$, the norm -dual of some E being a norm -space. All of these spaces are subspaces of the $L^0(\Omega ,\mathcal{F},\mathbb{P})$. The last space is the following one :

$$L^0(\Omega ,\mathcal{F},\mathbb{P})=\left\{X\right|X:\Omega \to \mathbb{R}\},$$

where ay X is $\mathcal{F}$-measurable. The probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is such that $\mathbb{P}$ is non -atomic and complete. All of the spaces are partially ordered by the pointwise partial ordering, namely $X\ge Y$, if and only if $X\left(\omega \right)\ge Y\left(\omega \right),\mathbb{P}$ almost surely. $E_{+}^{*}$ denotes the positive cone of any such space $E^{*}$ mentioned in this paper, namely $E_{+}^0=\{f\in E^{*}|f\left(X\right)\ge 0\}$ for any $X\in E_{+}$. $E_{+}$ is the cone of the $\mathbb{P}$ -almost surely positive elements of E. The Orlicz space $M^{\Phi }$, is the dual space of the Orlicz space $L^{\Psi }$, where $\Psi$ is some N-Young function and $\Phi ,\Psi$ are conjugate functions. Namely, $\Psi$ is a N-function, then ${\left(L^{\Psi }\right)}^{*}=M^{\Phi }$. Let us suppose that L is a non-trivial vector space. K is supposed to be a non-empty set of L and $K\ne \left\{0\right\}$. The real number $f\left(x\right)$ whenever mentioned below, is the value of $x\in L$ under some linear functional f. K is a wedge of L, if $K+K\subseteq K$, $\lambda ·K=\{\lambda ·k,k\in K\}\subseteq K$, for any $\lambda \in {\mathbb{R}}_{+}$. $t·x$ denotes the scalar product between any vector x of L and some real number t. K is a cone of L, if it is a wedge and moreover $K\cap (-K)=\left\{0\right\}$, where $0\in L$ is the zero element of L.

The Appendix contains some basic properties of Orlicz Spaces. About separation theorems and rest topological issues, the reader may look in [1].

The main issue of the paper is to prove the existence of strictly positive functionals in the case of the associate dual pairs of Orlicz Spaces. Strictly positive functionals are related to the definition of the Radon -Nikodym derivatives of some (local) martingale probability measure $\mathbb{Q}$ with respect to the nature probability measure $\mathbb{P}$. In general, strictly positive functionals are the pricing vectors in some commodity -price dual pair $\langle E,E^{*}\rangle$, where E is the commodity space and $E^{*}$ is the price space. The reader may look in [2] which is a seminal reference about related notions, which are in general common in Mathematical Finance. If E is a partially ordered linear space, whose positive cone is $E_{+}$ the set of strictly positive linear functionals is the following subset $E^{\prime }$ of $E^{*}$ : $f\in E^{\prime }$ if and only if $f\left(X\right)>0$ for any $X\in E_{+}\setminus \left\{0\right\}$.

1.1. Financial Framework

As it is considered in [3], we assume a ${\mathbb{R}}^{d+1}$- valued stochastic process, based on and adapted to the filtered probability space $(\Omega ,\mathcal{F},{\left(\mathcal{F}\right)}_{t\ge 0},\mathbb{P})$. The zero coordinate of this process $S^0$, denotes the risk -free asset and it is normalized to $S_t^0=1,t\ge 0$. S is locally bounded, namely there exists a sequence of stopping times ${\tau }_n,n\in \mathbb{N}$, increasing to ∞, such that the stopped processes $S_t^{{\tau }_n}=S_{t\wedge {\tau }_n}$ are uniformly bounded. This corresponds to the fact that S is assumed to be a c$\acute{a}$dl$\acute{a}$g process with uniformly bounded jumps. Also, a standrard assumption about the filtration is that it is right continuous, and ${\mathcal{F}}_0$ contains the null sets of ${\mathcal{F}}_{\infty }$. Also, a simple trading strategy is defined as follows -see also [3]:

Definition 1.1. 

A  simple trading strategy  H for a locally bounded stochastic process is an ${\mathbb{R}}^d$-valued stochastic process $H={\left(H_t\right)}_{t\ge 0}$, being of the form

$$H=\sum _{i=1}^n{h}_i{I}_{({\tau }_{i-1},{\tau }_i]},$$

where $0={\tau }_0\le {\tau }_1\le ...\le {\tau }_n<\infty$ are finite stopping times and $h_i$ are ${\mathcal{F}}_{{\tau }_{i-1}}$-measurable, ${\mathbb{R}}^d$ -valued functions. The associated  stochastic integral  $H·S$ is the stochastic process

$${(H·S)}_t=\sum _{i=1}^n(h_i,S_{{\tau }_i\wedge t}-S_{{\tau }_{i-1}\wedge t})=\sum _{i=1}^n\sum _{j=1}^d(h_i^j,S_{{\tau }_i\wedge t}^j-S_{{\tau }_{i-1}\wedge t}^j),0\le t\le \infty .$$

Its terminal value is the random variable

$${(H·S)}_{\infty }=\sum _{i=1}^n{h}_i(S_{{\tau }_i}-S_{{\tau }_{i-1}}).$$

Definition 1.2. 

H is  admissible, if the functions $h_i$ and stopped process $S^{{\tau }_n}$ are uniformly bounded.

For the next definition, see also [3]:

Definition 1.3. 

A probability measure $\mathbb{Q}$ on $\mathcal{F}$, which is equivalent (absolutely continuous) with respect to $\mathbb{P}$ is called an equivalent (absolutely continuous) local martingale measure, if S is a local martingale under $\mathbb{Q}$. We denote by ${\mathcal{M}}^e\left(S\right)$ the set of all such measures, and we say that S satisfies the condition of the existence of an equivalent local martingale measure (EMM), if ${\mathcal{M}}^e\left(S\right)\ne \varnothing$.

For the next Lemma, see also [3]:

Lemma 1.4.  

Let $\mathbb{Q}$ be a probability measure on $\mathcal{F}$, which is absolutely continuous with respect to $\mathbb{P}$. A locally bounded stochastic process S is a local martingale with respect to $\mathbb{Q}$, if and only if

$${\mathbb{E}}_{\mathbb{Q}}\left({(H·S)}_{\infty }\right)=0,$$

for any admissible simple trading strategy H.

Having this Lemma in mind, we define the following subspace of $L^{\infty }(\Omega ,\mathcal{F},\mathbb{P})$:

Definition 1.5. 

$$\begin{array}{c}\hfill K^s=\left\{{(H·S)}_{\infty }\right|H:simple,admissible\}\end{array}$$

and by $C^s$ the following convex cone of $L^{\infty }(\Omega ,\mathcal{F},\mathbb{P})$:

Definition 1.6. 

$C^s=K^s-L_{+}^{\infty }=\{f-k|f\in K^s,k\in L^{\infty },k\ge 0,\mathbb{P}-a.e.\}$

Compare the following Definition and [3]:

Definition 1.7. 

S satisfies theNo-Arbitragecondition ($N{A}^s$) with respect to simple trading strategies if

$$K^s\cap L_{+}^{\infty }(\Omega ,\mathcal{F},\mathbb{P})=\left\{0\right\},$$

(or equivalently

$$C^s\cap L_{+}^{\infty }(\Omega ,\mathcal{F},\mathbb{P})=\left\{0\right\}).$$

Also, compare the following Proposition and [3]:

Proposition 1.8.  

The condition (EMM) of xistence of an equivalent local martingale measure implies the condition ($N{A}^s$) of no-arbitrage with respect to simple trading strategies, but not vice versa.

Moreover, compare the following Definition and [3]. Also see [7]:

Definition 1.9.  

S satisfies the condition ofNo free lunch(NFL), if the closure $\overline{C}$ of $C^s$, taken with respect to the weak-star topology of $L^{\infty }(\Omega ,\mathcal{F},\mathbb{P})$ satisfies:

$$\overline{C}\cap L_{+}^{\infty }(\Omega ,\mathcal{F},\mathbb{P})=\left\{0\right\}.$$

The consequence of NFL Condition is the Kreps-Yan Theorem

(compare with [3]):

Theorem 1.10.  

A locally bounded stochastic process S satisfies the condition of (NFL), if and only if the condition (EMM) of the existence of an equivalent local martingale measure is satisfied.

The above Definitions of $K^s,C^s$, remind us of the space of stochastic processes mentioned in [5]. Specifically, the associated linear space $L_{sm}$ of the stochastic processs, mentioned above, may be replaced by the following linear space of stochastic processs:

$$x(t,\omega )=\sum _{i=0}^n{x}_i{I}_{E_i}\left(\omega \right)I_{\tau \left(\omega \right),T]}\left(t\right),$$

denotes the space of the simple, adapted processes on the propability space $(\Omega ,\mathcal{F},\mathbb{P})$. The meaning of x is that such a cash -stream pays $x_i$ if $E_i$ occurs, at the time ${\tau }_i$. The subspace $K^s$ is a natural generalization of the elements of $L_{sm}$ from the aspect of the extension of the time-horizon to the infinity and the extension of the payment elements $x_i$ on bounded random variables, in the sense of $L^{\infty }$. The above space odf stochastic processes may also considered to be the one, which arises in the case where $x_i$ lie in some Orlicz Space.

2. Results on the Strictly Positive Functionals in Orlicz Spaces

Theorem 2.1.  

Let Ψ is some N-function and $〈M^{\Phi },L^{\Psi }〉$ is the commodity-price duality, and K is the subspace of the replicated contingent claims, satisfying the property of No-Free Lunches:

$${\overline{(K-M_{+}^{\Phi })}}^{w^{*}}\cap M_{+}^{\Phi }=\left\{0\right\}.$$

This implies the existence of some $f_0\in L^{\Psi }$, such that $f_0\left(x\right)>0,x\in M_{+}^{\Phi }\setminus \left\{0\right\}$.

Proof.

$$\left\{f\in M^{\Phi }|f\in [0,\mathbf{1}]\right\}$$

is a weak-star compact set of $M^{\Phi }$ since it is convex, closed and bounded with respect to any norm being posed on $M^{\Phi }$, Let ${\delta }_n$ be a sequence of real numbers, such that $0<{\delta }_n<1$, and we apply the separation theorem between the convex sets $C={\overline{(K-M_{+}^{\Phi })}}^{w^{*}}$ and

$$B_{{\delta }_n}=\left\{f\in M^{\Phi }|\mathbb{E}\left(f\right)\ge {\delta }_n\right\}.$$

For these sets we have that $C\cap B_{{\delta }_n}=\varnothing$. Then there exists some $g_n\in L^{\Psi }$ such that

$$g_n\left(c\right)\le 0\le ϵ\le g_n\left(f\right),$$

where $f\in B_{{\delta }_n},c\in C$. If $a_n>0,n\in \mathbb{N}$ such that ${\sum }_{n=1}^{\infty }{a}_n=1$, while $\parallel g_n\parallel =b_n$, we take the functionals $\widetilde{g_n}={\displaystyle \frac{1}{b_n}}{g}_n$. If ${\sum }_{n=1}^{\infty }{\displaystyle \frac{a_n}{b_n}}<\infty$, then

$$\sum _{n=1}^{\infty }{a}_n\widetilde{g_n}=f_0.\square$$

The following Corollary arises from the fact that ${\left(L^{\Phi }\right)}^{*}=L^{\Phi }$, if $\Phi$ is both ${\Delta }_2$ and N function.

Corollary 2.2.  

Let Φ is some ${\Delta }_2$-function and $〈L^{\Phi },L^{\Phi }〉$ is the commodity-price duality, and K is the subspace of the replicated contingent claims, satisfying the property of No-Free Lunches:

$${\overline{(K-L_{+}^{\Phi })}}^{w^{*}}\cap L_{+}^{\Phi }=\left\{0\right\}.$$

This implies the existence of some $f_0\in L^{\Phi }$, such that $f_0\left(x\right)>0,x\in M_{+}^{\Phi }\setminus \left\{0\right\}$.

Proof.

$$\left\{f\in L^{\Phi }|f\in [0,\mathbf{1}]\right\}$$

is a weak-star compact set of $M^{\Phi }$ since it is convex, closed and bounded with respect to any norm being posed on $M^{\Phi }$, Let ${\delta }_n$ be a sequence of real numbers, such that $0<{\delta }_n<1$, and we apply the separation theorem between the convex sets $C={\overline{(K-L_{+}^{\Phi })}}^{w^{*}}$ and

$$B_{{\delta }_n}=\left\{f\in L^{\Phi }|\mathbb{E}\left(f\right)\ge {\delta }_n\right\}.$$

For these sets we have that

$$C\cap B_{{\delta }_n}=\varnothing .$$

Then there exists some $g_n\in L^{\Phi }$ such that

$$g_n\left(c\right)\le 0\le ϵ\le g_n\left(f\right),$$

where $f\in B_{{\delta }_n},c\in C$. If $a_n>0,n\in \mathbb{N}$ such that ${\sum }_{n=1}^{\infty }{a}_n=1$, while $\parallel g_n\parallel =b_n$, we take the functionals $\widetilde{g_n}={\displaystyle \frac{1}{b_n}}{g}_n$. If ${\sum }_{n=1}^{\infty }{\displaystyle \frac{a_n}{b_n}}<\infty$, then

$$\sum _{n=1}^{\infty }{a}_n\widetilde{g_n}=f_0.\square$$

Namely, we obtain the following theorem:

Theorem 2.3.  

Either Ψ is some N-function and $〈M^{\Phi },L^{\Psi }〉$ is the commodity-price duality, or Φ is some ${\Delta }_2$-function and $〈L^{\Phi },L^{\Phi }〉$ is the commodity-price duality, the existence of $f_0$ implies the existence of a probability measure $\mathbb{Q}$ such that ${\displaystyle \frac{d\mathbb{Q}}{d\mathbb{P}}}=f_0$, such that $C\cap E_{+}^{*}=\left\{0\right\}$, where $C={\overline{(K-E_{+}^{*})}}^{\sigma (E^{*},E)}$, $E^{*}=M^{\Phi }$ if Ψ is some N-function, or $E^{*}=L^{\Phi }$, if Φ is some ${\Delta }_2$-function. Conversely, given an equivalent probability measure to $\mathbb{P}$, defined on $(\Omega ,\mathcal{F})$, such that

$${\displaystyle \frac{d\mathbb{Q}}{d\mathbb{P}}}\in E_{+}\setminus \left\{0\right\},$$

where $E=L^{\Psi }$ if Ψ is some N-function, or $E=L^{\Phi }$, if Φ is some ${\Delta }_2$-function, then $C=\{x\in E|{\mathbb{E}}_{\mathbb{Q}}\left(x\right)\le 0\}$ is $\sigma (E^{*},E)$-closed and satisfies $C\cap E_{+}^{*}=\left\{0\right\}$.

Proof. Let $(\Omega ,\mathcal{F},\mathbb{P})$ be the initial probability space. We define:

$$\mathbb{Q}\left(A\right)={\int }_A{f}_{0}d\mathbb{P},A\in \mathcal{F}.$$

The converse is alike to the [3]. □

The class of Orlicz spaces $\left\{L^{\Phi }\right\}$, whose quasi-interior of the cone $L_{+}^{\Phi }$ in a normed linear space $L^{\Phi }$ is non-empty, coincides with the class of the spaces $\left\{L^{\Phi }\right\}$, such that the space $L^{\Phi }$ is separable, the cone $L_{+}^{\Phi }$ is closed and holds ${\left(L^{\Phi }\right)}^{*}=L^{\Phi }$. If $\Phi$ obtains the ${\Delta }_2$-property, then ${\left(L^{\Phi }\right)}^{*}=L^{\Phi }$ (see [8]). Some $x\in L_{+}$, where L is a partially ordered linear space by the cone $L_{+}$ is a quasi-interior point of the cone $L_{+}$ if $L={\overline{I}}_x$, where closure is obtained by the norm topology and $I_x={\cup }_{n=1}^{\{}infty][-n·x,n·x]$. The order interval $[a,b]$ is actually the subset $(b-L_{+})\cap (a+L_{+})$, where $a,b\in L$. By $t·y$, we denote the scalar product between $y\in L$ and $t\in \mathbb{R}$. The subspace $I_x$ of L defined in the associated way is usually called the solid suspace of L generated by x.

These spaces are reflexive if $\Phi \in {\Delta }_2\cap {\nabla }_2$, and the above theorems are valid for weak topology, since weak star and weak topology coincide. This reflexivity point is of particular interest, since it provides the above difference between the dual pairs. The entire properties of Orlicz Spaces are presented in [10],

3. Some Further Implications of Study

The following two propositions imply a relation between the Orlicz Spaces are the $L^p(\Omega ,\mathcal{F},\mathbb{P})$ spaces.

Proposition 3.1.  

We suppose that Φ is a Young function, such that $L^{\Phi }\ne \left\{0\right\}$. Then, if $X\in L_{+}^{\Phi }\setminus \left\{0\right\}$, then $X\in L^1{(\Omega ,\mathcal{F},\mathbb{P})}_{+}\setminus \left\{0\right\}$.

Proof. 

If $X\in L_{+}^{\Phi }$, then by applying Jensen’s Inequality, $\Phi \left(\mathbb{E}\right(X\left)\right)\le \mathbb{E}(\Phi (X\left)\right)<\infty$. Thus, $\mathbb{E}\left(X\right)<\infty$, because since there exists some sequence ${\left(X_n\right)}_{n\in \mathbb{N}}$ of step functions, such that $\mathbb{E}\left(X_n\right)<\infty$ and $X_n\to X,\mathbb{P}-a.e.$ pointwise, if $a_n={lim}_{n\in \mathbb{N}}\mathbb{E}\left(X_n\right)=+\infty$, then since $\Phi$ is a Young function, we would have that ${lim}_{n\in \mathbb{N}}\Phi \left(a_n\right)=+\infty$. But in this case, the Jensen’s Inequality is violated, which is a contradiction. □

The Definition of an Orlicz Heart $M^{\Phi }$ is mentioned in the Appendix. It is in general a convex set. Any such set generates the following cone :

Definition 3.2.  

Since $M^{\Phi }$ is a convex set of $L^{\Phi }$.

$$K^{\Phi }=cone\left(M^{\Phi }\right):=\{t·X|X\in M^{\Phi },t\in {\mathbb{R}}_{+}\},$$

is called  cone generated by  $M^{\Phi }$.

Proposition 3.3.  

If $X\in L_{+}^{\Phi }\setminus \left\{0\right\}$, then ${\Phi (\parallel X\parallel }_1{)\le \parallel X\parallel }_{\Phi }$, if we consider that $L^{\Phi }$ is a normed linear space, where ${\parallel .\parallel }_{\Phi }$ is the Luxemburg norm and $M^{\Phi }$ is the corresponding Orlicz Heart.

Proof. 

Since $X\in M^{\Phi }$, then for any $c>0$, $\mathbb{E}(\Phi (c·X\left)\right)<\infty$. Then the set

$$\{\lambda >0:{\mathbb{E}}_{\mathbb{P}}\left[\Phi \left({\displaystyle \frac{X}{\lambda }}\right)\right]\le 1\},$$

is non-empty. □

As a mattet of fact, the cones of the Orlicz Spaces which are naturally asscociated with them through the pointwise partial ordering, may be considered as cones of admissible contingent claims in some $L^p(\Omega ,\mathcal{F},\mathbb{P})$ space. This kind of embedding may be studied in a more detailed way, associated with the No- Arbitrage and No free lunch properties. This may be considered as a direction of further study in this topic.