IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2409.05194.html
   My bibliography  Save this paper

Risk measures on incomplete markets: a new non-solid paradigm

Author

Listed:
  • Vasily Melnikov

Abstract

The abstract theory of risk measures is well-developed for certain classes of solid subspaces of $L^{0}$. We provide an example to illustrate that this framework is insufficient to deal with the subtleties of incomplete markets. To remedy this problem, we consider risk measures on the subspace generated by a closed, absolutely convex, and bounded subset $K\subset L^{0}$, which represents the attainable securities. In this context, we introduce the equicontinuous Fatou property. Under the existence of a certain topology $\tau$ on $\mathrm{span}(K)$, interpreted as a generalized weak-star topology, we obtain an equivalence between the equicontinuous Fatou property, and lower semicontinuity with respect to $\tau$. As a corollary, we obtain tractable dual representations for such risk measures, which subsumes essentially all known results on weak-star representations of risk measures. This dual representation allows one to prove that all risk measures of this form extend, in a maximal way, to the ideal generated by $\mathrm{span}(K)$ while preserving a Fatou-like property.

Suggested Citation

  • Vasily Melnikov, 2024. "Risk measures on incomplete markets: a new non-solid paradigm," Papers 2409.05194, arXiv.org.
  • Handle: RePEc:arx:papers:2409.05194
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2409.05194
    File Function: Latest version
    Download Restriction: no
    ---><---

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2409.05194. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.