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# Maximum Lebesgue Extension Of Convex Risk Measures

## Author

Listed:
• Keita Owari

(Graduate School of Economics, University of Tokyo, Tokyo)

## Abstract

Given a convex risk measure on $L^\infty$ having the Lebesgue property, we construct a solid space of random variables on which the original risk measure is extended preserving the Lebesgue property (on the entire space). This space is an order-continuous Banach lattice, and is maximum among all solid spaces admitting such a regular extension. We then characterize the space in terms of uniform integrability of certain families. As a byproduct, we present a generalization of Jouini-Schachermayer-Touziâ€™s theorem on the weakcompactness characterization of Lebesgue property, which is valid for any solid vector spaces of random variables, and does not require any topological property of the space.

## Suggested Citation

• Keita Owari, 2012. "Maximum Lebesgue Extension Of Convex Risk Measures," CARF F-Series CARF-F-287, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
• Handle: RePEc:cfi:fseres:cf287
as

File URL: http://www.carf.e.u-tokyo.ac.jp/pdf/workingpaper/fseries/299.pdf

## References listed on IDEAS

as
1. Freddy Delbaen, 2009. "Risk Measures For Non-Integrable Random Variables," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 329-333.
2. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
3. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2005. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, vol. 9(3), pages 369-387, July.
4. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214.
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