# Compactness, Optimality and Risk

Orihuela, Jose

### Abstract

We propose a tour around the following:

\textbf{Theorem:} Let *TeX Embedding failed!* be a coercive, proper, convex and lower semicontinuous function on the real Banach space *TeX Embedding failed!*. They are equivalent:

\begin{enumerate}

\item *TeX Embedding failed!* \item *TeX Embedding failed!*

\end{enumerate}

When the domain of *TeX Embedding failed!* has interior point the former conditions are equivalent to the reflexivity of the Banach space *TeX Embedding failed!*.

For separable Banach spaces I obtained the result for maps with bounded domain answering a question by Jouini, Schachermayer and Touizi. The proof is included in their paper for the study of risk measures with the Lebesgue property, [5]. The same theorem for risk measures has been recently proved by Delbaen [2] using the original version of James compactness Theorem together with a homogenisation trick for arbitrary spaces of the form

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where *TeX Embedding failed!* is a probability space. The result when the domain of *TeX Embedding failed!* has interior point was obtained by Calvert and Fitzpatrick [1,3]. Stephen Simons noticed that there were omissions in [3] making their results false. The erratum [1] makes [3] more difficult to follow. Indeed the main addendum is necessary to correct non written proofs of lemmas in [3] which are adapted from [5]. Our approach is based on extensions of Simons Suplimsup Theorem of [7] for unbounded domains, together with a perturbed James Theorem proved in [6]. For instance, it provides us the proof of *TeX Embedding failed!* if the map *TeX Embedding failed!* is not assumed to be coercive but the Banach space *TeX Embedding failed!* has a *TeX Embedding failed!*-sequentially compact dual unit ball *TeX Embedding failed!*. More applications to convex risk measures were presented in the talk.

References:

1. B. Calvert and S. Fitzpatrick, *TeX Embedding failed!* Math. Zeitschrift *TeX Embedding failed!* (2000) 627.

2. F. Delbaen, *TeX Embedding failed!* Optimality and Risk- Modern Trends in Mathematical Finance Springer 2009, 39--48.

3. S. Fitzpatrick and B. Calvert *TeX Embedding failed!* Math. Zeitschhrift *TeX Embedding failed!* (1985) 555--560. \item R.C. James, \textsl{Reflexivity and the sup of linear functionals} Isr. J. Math. *TeX Embedding failed!* (1972) 289--300.

4. E. Jouini, W. Schachermayer and N. Touizi, *TeX Embedding failed!*, Advances in Mathematical Economics, Springer 2006, *TeX Embedding failed!* 49--71.

5. M. Ruiz Gal\'{a}n and S. Simons, *TeX Embedding failed!*, Bull. Australian Math. Soc. *TeX Embedding failed!* (2002) 43--56.

6. S. Simons, *TeX Embedding failed!*, Pacific J. Math. *TeX Embedding failed!* (1972) 703--708.

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