# An Application of Maximal Exponential Models to Duality Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Orlicz Spaces and Maximal Exponential Models

- (i)
- $\mathrm{\Phi}(0)=0$,
- (ii)
- ${lim}_{x\to \infty}\mathrm{\Phi}(x)=+\infty $,
- (iii)
- $\mathrm{\Phi}(x)<+\infty $ in a neighborhood of 0.

**Definition**

**1.**

- 1.
- $p(\theta )\propto {p}^{(1-\theta )}{q}^{\theta}\in \mathcal{P}$, $\forall \theta \in I$;
- 2.
- $p(\theta )\propto {e}^{\theta u}p\in \mathcal{P}$, $\forall \theta \in I$, where $u\in {L}^{{\mathrm{\Phi}}_{1}}(p)$ and $p(0)=p,\phantom{\rule{4pt}{0ex}}p(1)=q$.

**Definition**

**2.**

**Definition**

**3.**

**Proposition**

**1.**

**Proposition**

**2.**

**Theorem**

**1.**

- (i)
- $q\in \mathcal{E}(p)$;
- (ii)
- q is connected to p by an open exponential arc;
- (iii)
- $\mathcal{E}(p)=\mathcal{E}(q)$;
- (iv)
- $log\frac{q}{p}\in {L}^{{\mathrm{\Phi}}_{1}}(p)\cap {L}^{{\mathrm{\Phi}}_{1}}(q);$
- (v)
- ${L}^{{\mathrm{\Phi}}_{1}}(p)={L}^{{\mathrm{\Phi}}_{1}}(q)$;
- (vi)
- $\frac{q}{p}\in {L}^{1+\epsilon}(p)$ and $\frac{p}{q}\in {L}^{1+\epsilon}(q),$ for some $\epsilon >0$;
- (vii)
- the
`mixture transport mapping`$$\begin{array}{cc}\hfill {}^{m}{\mathbb{U}}_{p}^{q}:{L}^{{\mathrm{\Psi}}_{1}}(p)& \u27f6{L}^{{\mathrm{\Psi}}_{1}}(q)\hfill \\ \hfill v& \mapsto \frac{p}{q}v,\hfill \end{array}$$

**Corollary**

**1.**

**Proposition**

**3.**

## 3. Duality Results

**Proposition**

**4.**

**Theorem**

**2.**

**Proof.**

## 4. Financial Application

**Definition**

**4.**

**Proposition**

**5.**

**Theorem**

**3.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Santacroce, M.; Siri, P.; Trivellato, B. An Application of Maximal Exponential Models to Duality Theory. *Entropy* **2018**, *20*, 495.
https://doi.org/10.3390/e20070495

**AMA Style**

Santacroce M, Siri P, Trivellato B. An Application of Maximal Exponential Models to Duality Theory. *Entropy*. 2018; 20(7):495.
https://doi.org/10.3390/e20070495

**Chicago/Turabian Style**

Santacroce, Marina, Paola Siri, and Barbara Trivellato. 2018. "An Application of Maximal Exponential Models to Duality Theory" *Entropy* 20, no. 7: 495.
https://doi.org/10.3390/e20070495