# 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

- Pistone, G.; Sempi, C. An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat.
**1995**, 23, 1543–1561. [Google Scholar] [CrossRef] - Vigelis, R.F.; Cavalcante, C.C. On φ-families of probability distributions. J. Theor. Prob.
**2013**, 26, 870–884. [Google Scholar] [CrossRef] - De Andrade, L.H.F.; Vieira, F.L.J.; Vigelis, R.F.; Cavalcante, C.C. Mixture and Exponential Arcs on Generalized Statistical Manifold. Entropy
**2018**, 20, 147. [Google Scholar] [CrossRef] - Imparato, D.; Trivellato, B. Geometry of Extendend Exponential Models. In Algebraic and Geometric Methods in Statistics; Cambridge University Press: Cambridge, UK, 2009; pp. 307–326. [Google Scholar]
- Cena, A.; Pistone, G. Exponential Statistical Manifold. Ann. Inst. Stat. Math.
**2007**, 59, 27–56. [Google Scholar] [CrossRef] - Santacroce, M.; Siri, P.; Trivellato, B. New results on mixture and exponential models by Orlicz spaces. Bernoulli
**2016**, 22, 1431–1447. [Google Scholar] [CrossRef] [Green Version] - Santacroce, M.; Siri, P.; Trivellato, B. Exponential models by Orlicz spaces and Applications. Bernoulli
**2017**, in press. [Google Scholar] - Brigo, D.; Pistone, G. Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds. arXiv
**2016**, arXiv:1601.04189v. [Google Scholar] - Lods, B.; Pistone, G. Information geometry formalism for the spatially homogeneous Boltzmann equation. Entropy
**2015**, 17, 4323–4363. [Google Scholar] [CrossRef] - Santacroce, M.; Siri, P.; Trivellato, B. On Mixture and Exponential Connection by Open Arcs. In Geometric Science of Information; Nielsen, F., Barbaresco, F., Eds.; Springer International Publishing: Cham, Switzerland, 2017; pp. 577–584. ISBN 978-3-319-68444-4. [Google Scholar]
- Pistone, G. Examples of the application of nonparametric information geometry to statistical physics. Entropy
**2013**, 15, 4042–4065. [Google Scholar] [CrossRef] - Biagini, S.; Frittelli, M. Utility maximization in incomplete markets for unboundee processes. Financ. Stoch.
**2005**, 9, 493–517. [Google Scholar] [CrossRef] - Schachermayer, W. Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Prob.
**2001**, 11, 694–734. [Google Scholar] [CrossRef] [Green Version] - Frittelli, M. The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Financ.
**2000**, 10, 39–52. [Google Scholar] [CrossRef] - Delbaen, F.; Grandits, P.; Rheinländer, T.; Samperi, D.; Schweizer, M.; Stricker, C. Exponential hedging and entropic penalties. Math. Financ.
**2002**, 12, 99–123. [Google Scholar] [CrossRef] - Nock, R.; Magdalou, B.; Briys, E.; Nielsen, F. Mining Matrix Data with Bregman Matrix Divergences for Portfolio Selection. In MAtrix Information Geometry; Nielsen, F., Bathia, R., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 373–402. [Google Scholar]
- Nock, R.; Magdalou, B.; Briys, E.; Nielsen, F. On tracking portfolios with certainty equivalents on a generalization of Markowitz model: The fool, the wise and the adaptive. In Proceedings of the 28th International Conference on Machine Learning, Bellevue, WA, USA, 28 June–2 July 2011; Omnipress: Madison, WI, USA, 2011; pp. 73–80. [Google Scholar]
- Rodrigues, A.F.P.; Cavalcante, C.C. Principal Curves for Statistical Divergences and an Application to Finance. Entropy
**2018**, 20, 333. [Google Scholar] [CrossRef] - Rodrigues, A.F.P.; Guerreiro, M.; Cavalcante, C.C. Deformed Exponentials and Portfolio Selection. Int. J. Mod. Phys. C
**2018**, 29. [Google Scholar] [CrossRef]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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