×

zbMATH — the first resource for mathematics

Uncertainty modelling and conditioning with convex imprecise previsions. (English) Zbl 1123.62077
Summary: Two classes of imprecise previsions, which we termed convex and centered convex previsions, are studied in this paper in a framework close to P. Walley’s [Statistical reasoning with imprecise probabilities. (1991; Zbl 0732.62004)] and P. M. Williams’ [Notes on conditional previsions. Res. Rep. School Math. Phys. Sci., Univ. Sussex (1975)] theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural extension, which is useful in correcting a large class of inconsistent imprecise probability assessments, characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. We discuss in particular their usage in (financial) risk measurement. In a final part we introduce convex imprecise previsions in a conditional environment and investigate their basic properties, showing how several of the preceding notions may be extended and the way the generalised Bayes rule applies.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Artzner, P., Application of coherent risk measures to capital requirements in insurance, North American actuarial journal, 3, 11-26, (1999) · Zbl 1082.91525
[2] Artzner, P.; Delbaen, F.; Eber, S.; Heath, D., Coherent measures of risk, Mathematical finance, 9, 203-228, (1999) · Zbl 0980.91042
[3] de Cooman, G.; Aeyels, D., Supremum preserving upper probabilities, Information sciences, 118, 173-212, (1999) · Zbl 0952.60009
[4] Cozman, F.G., Computing posterior upper expectations, International journal of approximate reasoning, 24, 191-205, (2000) · Zbl 0995.60005
[5] Crisma, L., Lezioni di calcolo delle probabilità, (1998), Edizioni Goliardiche
[6] Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Finance and stochastics, 6, 429-447, (2002) · Zbl 1041.91039
[7] Föllmer, H.; Schied, A., Robust preferences and convex measures of risk, (), 39-56 · Zbl 1022.91045
[8] Jouini, E.; Kallal, H., Viability and equilibrium in securities markets with frictions, Mathematical finance, 9, 275-292, (1999) · Zbl 0980.91022
[9] Klir, G.J.; Wierman, M.J., Uncertainty-based information, (1998), Physica-Verlag Heidelberg · Zbl 0902.68061
[10] Maaß, S., Exact functionals and their core, Statistical paper, 43, 75-93, (2002) · Zbl 1038.46005
[11] Nau, R.F., Indeterminate probabilities on finite sets, The annals of statistics, 20, 1737-1767, (1992) · Zbl 0782.62006
[12] Pelessoni, R.; Vicig, P., Convex imprecise previsions, Reliable computing, 9, 465-485, (2003) · Zbl 1037.60003
[13] Pelessoni, R.; Vicig, P., Imprecise previsions for risk measurement, International journal of uncertainty, fuzziness and knowledge-based systems, 11, 393-412, (2003) · Zbl 1074.91030
[14] R. Pelessoni, P. Vicig, Convex imprecise previsions: basic issues and applications, in: J.M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proc. ISIPTA’03, Carleton Scientific, 2003, pp. 423-436 · Zbl 1037.60003
[15] Shafer, G.; Gillett, P.R.; Scherl, R.B., A new understanding of subjective probability and its generalization to lower and upper prevision, International journal of approximate reasoning, 33, 1-49, (2003) · Zbl 1092.68098
[16] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall · Zbl 0732.62004
[17] Walley, P.; Pelessoni, R.; Vicig, P., Direct algorithms for checking consistency and making inferences from conditional probability assessments, Journal of statistical planning and inference, 126, 119-151, (2004) · Zbl 1075.62002
[18] K. Weichselberger, T. Augustin, On the symbiosis of two concepts of conditional interval probability, in: J.M. Bernard, T. Seidenfeld, M. Zaffalon (Eds)., Proc. ISIPTA’03, Carleton Scientificic, 2003, pp. 608-629
[19] P.M. Williams, Notes on conditional previsions. Research Report, School of Mathematics and Physical Science, University of Sussex, 1975 · Zbl 1114.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.