zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Recent progress in random metric theory and its applications to conditional risk measures. (English) Zbl 1238.46058
Summary: The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}$-convex topology and in particular a characterization for a locally $L^{0}$-convex module to be $L^{0}$-pre-barreled. Section 7 gives some basic results on $L^{0}$-convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}$-type of conditional convex risk measure and every continuous $L^{p}$-type of convex conditional risk measure $(1 \leq p < +\infty)$ can be extended to an $L_\mathcal{F}^\infty \left( \mathcal{E} \right)$-type of $\sigma_{\varepsilon ,\lambda} \left( {L_\mathcal{F}^\infty \left( \mathcal{E} \right),L_\mathcal{F}^1 \left( \mathcal{E} \right)} \right)$-lower semicontinuous conditional convex risk measure and an $L_\mathcal{F}^{p} \left( \mathcal{E} \right)$-type of $\mathcal{T}_{\varepsilon ,\lambda }$-continuous conditional convex risk measure $(1 \leq p < +\infty)$, respectively.

MSC:
 46S50 Functional analysis in probabilistic metric linear spaces 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46A25 Reflexivity and semi-reflexivity of topological linear spaces 46H25 Normed modules and Banach modules, topological modules 47H40 Random operators (nonlinear) 52A41 Convex functions and convex programs (convex geometry) 91B16 Utility theory 91B30 Risk theory, insurance 91B70 Stochastic models in economics
Full Text:
References:
 [1] Artzner P, Delbaen F, Eber J M, et al. Coherent measures of risk. Math Finance, 1999, 9: 203--228 · Zbl 0980.91042 · doi:10.1111/1467-9965.00068 [2] Berberian S K. Lectures in Functional Analysis and Operator Theory. New York: Springer-Verlag, 1974 · Zbl 0296.46002 [3] Biagini S, Frittelli M. On continuity properties and dual representation of convex and monotone functionals on Frechet lattices. Working paper, 2006 [4] Bion-Nadal J. Conditional risk measures and robust representation of convex conditional risk measures. CMAP Preprint, 557, 2004 [5] Breckner W W, Scheiber E. A Hahn-Banach extension theorem for linear mappings into ordered modules. Mathematica, 1977, 19: 13--27 · Zbl 0396.46002 [6] Cheridito P, Delbaen F, Eber J M, et al. Coherent multiperiod risk adjusted values and Bellman’s principle. Ann Oper Res, 2007, 152: 5--22 · Zbl 1132.91484 · doi:10.1007/s10479-006-0132-6 [7] Cheridito P, Delbaen F, Kupper M. Dynamic monetary risk measures for bounded discrete-time processes. Electron J Probab, 2006, 11: 57--106 · Zbl 1184.91109 [8] Cheridito P, Li T. Dual characterization of properties of risk measures on Orlicz hearts. Math Finance Econ, 2008, 2: 29--55 · Zbl 1181.91092 · doi:10.1007/s11579-008-0013-7 [9] Delbaen F. Coherent risk measures. Cattedra Galileiana, 2000 · Zbl 1320.91066 [10] Delbaen F. Coherent risk measures on general probability spaces. In: Sandmann K, Schönbucher P J, eds. Advances in Finance and Stochastics. Berlin: Springer-Verlag, 2002, 1--37 · Zbl 1020.91032 [11] Detlefsen K, Scandolo G. Conditional and dynamic convex risk measures. Finance Stoch, 2005, 9: 539--561 · Zbl 1092.91017 · doi:10.1007/s00780-005-0159-6 [12] Diestel J, Uhl Jr J J. Vector Measures. Math Surveys, No. 15. Providence, RI: Amer Math Soc, 1977 [13] Dunford N, Schwartz J T. Linear Operators (I). New York: Interscience, 1957 · Zbl 0084.10402 [14] Ekeland I, Témam R. Convex Analysis and Variational Problems, Chapter (I). Philadelphia: SIAM, 1999 [15] Filipović D, Kupper M, Vogelpoth N. Separation and duality in locally L 0-convex modules. J Funct Anal, 2009, 256: 3996--4029 · Zbl 1180.46055 · doi:10.1016/j.jfa.2008.11.015 [16] Filipović D, Kupper M, Vogelpoth N. Approaches to conditional risk. Working paper series No. 28, Vienna Institute of Finance, 2009 · Zbl 1255.91178 [17] Filipović D, Svindland G. Convex risk measures beyond bounded risks, or the canonical model space for law-invariant convex risk measures is L 1. Working paper series No. 2, Vienna Institute of Finance, 2008 · Zbl 1278.91086 [18] Föllmer H, Penner I. Convex risk measures and the dynamics of their penalty functions. Statist Decisions, 2006, 24: 61--96 · Zbl 1186.91119 [19] Föllmer H, Schied A. Convex measures of risk and trading constraints. Finance Stoch, 2002, 6: 429--447 · Zbl 1041.91039 · doi:10.1007/s007800200072 [20] Föllmer H, Schied A. Robust preferences and convex measures of risk. In: Sandmann K, Schönbucher P J, eds. Advances in Finance and Stochastics. Berlin: Springer-Verlag, 2002, 39--56 · Zbl 1022.91045 [21] Föllmer H, Schied A. Stochastic Finance, An Introduction in Discrete Time. Berlin-New York: De Gruyter, 2002 · Zbl 1125.91053 [22] Frittelli M, Rosazza Gianin E. Putting order in risk measures. J Bank Finance, 2002, 26: 1473--1486 · doi:10.1016/S0378-4266(02)00270-4 [23] Frittelli M, Rosazza Gianin E. Dynamic convex risk measures. In: Szegö G, ed. New Risk Measures for the 21st Century. New York: John Wiley & Sons, 2004, 227--248 [24] Guo T X. The theory of probabilistic metric spaces with applications to random functional analysis. Master’s thesis. Xi’an: Xi’an Jiaotong University, 1989 [25] Guo T X. Random metric theory and its applications. PhD thesis. Xi’an: Xi’an Jiaotong University, 1992 [26] Guo T X. Extension theorems of continuous random linear operators on random domains. J Math Anal Appl, 1995, 193: 15--27 · Zbl 0879.47018 · doi:10.1006/jmaa.1995.1221 [27] Guo T X. The Radon-Nikodým property of conjugate spaces and the w*-equivalence theorem for w*-measurable functions. Sci China Ser A, 1996, 39: 1034--1041 · Zbl 0868.46014 [28] Guo T X. Module homomorphisms on random normed modules. Northeast Math J, 1996, 12: 102--114 · Zbl 0858.60012 [29] Guo T X. Random duality. Xiamen Daxue Xuebao Ziran Kexue Ban, 1997, 36: 167--170 [30] Guo T X. A characterization for a complete random normed module to be random reflexive. Xiamen Daxue Xuebao Ziran Kexue Ban, 1997, 36: 499--502 · Zbl 0902.46053 [31] Guo T X. Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal Funct Appl, 1999, 1: 160--184 · Zbl 0965.46010 [32] Guo T X. Representation theorems of the dual of Lebesgue-Bochner function spaces. Sci China Ser A, 2000, 43: 234--243 · Zbl 0959.46024 · doi:10.1007/BF02897846 [33] Guo T X. Survey of recent developments of random metric theory and its applications in China (I). Acta Anal Funct Appl, 2001, 3: 129--158 · Zbl 0989.54035 [34] Guo T X. Survey of recent developments of random metric theory and its applications in China (II). Acta Anal Funct Appl, 2001, 3: 208--230 · Zbl 0989.54036 [35] Guo T X. The theory of random normed modules and its applications. In: Liu P D, ed. Proceedings of International Conference & 13th Academic Symposium in China on Functional Space Theory and Its applications. London: Research Information Ltd UK, 2004, 57--66 [36] Guo T X. Several applications of the theory of random conjugate spaces to measurability problems. Sci China Ser A, 2007, 50: 737--747 · Zbl 1128.46031 · doi:10.1007/s11425-007-0023-6 [37] Guo T X. The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure. Sci China Ser A, 2008, 51: 1651--1663 · Zbl 1167.46049 · doi:10.1007/s11425-008-0047-6 [38] Guo T X. A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module. arXiv: 0908.1843 [39] Guo T X. Relations between some basic results derived from two kinds of topologies for a random locally convex module. J Funct Anal, 2010, 258: 3024--3047 · Zbl 1198.46058 · doi:10.1016/j.jfa.2010.02.002 [40] Guo T X. The theory of module homomorphisms in complete random inner product modules and its applications to Skorohod’s random operator theory. Submitted [41] Guo T X, Chen X X. Random duality. Sci China Ser A, 2009, 52: 2084--2098 · Zbl 1193.46048 · doi:10.1007/s11425-009-0149-9 [42] Guo T X, Li S B. The James theorem in complete random normed modules. J Math Anal Appl, 2005, 308: 257--265 · Zbl 1077.46061 · doi:10.1016/j.jmaa.2005.01.024 [43] Guo T X, Peng S L. A characterization for an L({$\mu$},K)-topological module to admit enough canonical module homomorphisms. J Math Anal Appl, 2001, 263: 580--599 · Zbl 1014.46018 · doi:10.1006/jmaa.2001.7637 [44] Guo T X, Xiao H X. A separation theorem in random normed modules. Xiamen Daxue Xuebao Ziran Kexue Ban, 2003, 42: 270--274 · Zbl 1046.46020 [45] Guo T X, Xiao H X, Chen X X. A basic strict separation theorem in random locally convex modules. Nonlinear Anal, 2009, 71: 3794--3804 · Zbl 1184.46068 · doi:10.1016/j.na.2009.02.038 [46] Guo T X, You Z Y. The Riesz’s representation theorem in complete random inner product modules and its applications. Chinese Ann Math Ser A, 1996, 17: 361--364 · Zbl 0940.60067 [47] Guo T X, You Z Y. A note on pointwise best approximation. J Approx Theory, 1998, 93: 344--347 · Zbl 0912.41016 · doi:10.1006/jath.1997.3173 [48] Guo T X, Zeng X L. Existence of continuous nontrivial linear functionals on random normed modules. Chinese J Engrg Math, 2008, 25: 117--123 · Zbl 1164.46340 [49] Guo T X, Zeng X L. Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal, 2010, 73: 1239--1263 · Zbl 1202.46055 · doi:10.1016/j.na.2010.04.050 [50] Guo T X, Zhao S E, Zeng X L. On the analytic foundation of the module approach to conditional risk. Submitted [51] Guo T X, Zhu L H. A characterization of continuous module homomorphisms on random seminormed modules and its applications. Acta Math Sin (Engl Ser), 2003, 19: 201--208 · Zbl 1027.60069 · doi:10.1007/s10114-002-0210-x [52] He S W, Wang J G, Yan J A. Semimartingales and Stochastic Analysis. Beijing: Science Press, 1995 [53] Kaina M, Rüschendorf L. On convex risk measures on L p. Working paper, 2007 [54] Kantorovic L V. The method of successive approximations for functional equations. Acta Math, 1939, 71: 63--97 · Zbl 65.0520.02 · doi:10.1007/BF02547750 [55] Krätschmer V. On {$\sigma$}-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model. SFB 649 discussion paper 2007-010. Berlin: Humboldt University, 2007 [56] Kupper M, Vogelpoth N. Complete L 0-normed modules and automatic continuity of monotone convex functions. Working paper series No. 10, Vienna Institute of Finance, 2008 [57] Neveu J. Mathematical Foundations of the Calculus of Probabilities. San Francisco: Holden Day, 1965 · Zbl 0137.11301 [58] Peng S. Nonlinear expectations, nonlinear evaluations and risk measures. In: Lecture notes in Mathematics 1856. New York: Springer, 2004, 165--253 · Zbl 1127.91032 [59] Rockafellar R T. Conjugate duality and optimization. Regional Conference Series in Applied Mathematics, Vol. 16. Philadelphia: SIAM, 1974 · Zbl 0296.90036 [60] Rosazza Gianin E. Risk measures via g-expectations. Insurance Math Econom, 2006, 39: 19--34 · Zbl 1147.91346 · doi:10.1016/j.insmatheco.2006.01.002 [61] Ruszczyński A, Shapiro A. Optimization of convex risk functions. Math Opers Res, 2006, 31: 433--452 · Zbl 1181.90281 · doi:10.1287/moor.1050.0186 [62] Schachermayer W. A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance Math Econom, 1992, 11: 249--257 · Zbl 0781.90010 · doi:10.1016/0167-6687(92)90013-2 [63] Schweizer B, Sklar A. Probabilistic Metric Spaces. New York: Elsevier, 1983; reissued by New York: Dover Publications, 2005 · Zbl 0546.60010 [64] Song Y, Yan J A. The representations of two types of functionals on L {$\Omega$},F) and L{$\Omega$},F, P). Sci China Ser A, 2006, 49: 1376--1382 · Zbl 1177.46020 · doi:10.1007/s11425-006-2010-8 [65] Song Y, Yan J A. Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders. Insurance Math Econom, 2009, 45: 459--465 · Zbl 1231.91237 · doi:10.1016/j.insmatheco.2009.09.011 [66] Song Y, Yan J A. An overview of representation theorems for static risk measures. Sci China Ser A, 2009, 52: 1412--1422 · Zbl 1184.91114 · doi:10.1007/s11425-009-0122-7 [67] Vuza D. The Hahn-Banach theorem for modules over ordered rings. Rev Roumaine Math Pures Appl, 1982, 9: 989--995 · Zbl 0505.06010 [68] You Z Y, Guo T X. Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in L p({$\mu$},X). J Approx Theory, 1994, 78: 314--320 · Zbl 0808.41024 · doi:10.1006/jath.1994.1081 [69] Zowe J. A duality theorem for a convex programming problem in order complete lattices. J Math Anal Appl, 1975, 50: 273--287 · Zbl 0314.90079 · doi:10.1016/0022-247X(75)90022-0