Compactness, Optimality and Risk

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

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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