Уважаемые авторы, просим обратить внимание!
Подача статьи осуществляется только через личный кабинет электронной редакции.
DOI: 10.23671/VNC.2019.21.44629
Булевозначный подход к анализу условного риска
Сапата Х. М.
Владикавказский математический журнал. 2019. Том 21. Выпуск 4.С.71-89.
Аннотация: С помощью методов булевозначного анализа устанавливается принцип переноса между теорией двойственности классической выпуклой меры риска и теорией двойственности меры условного риска. А именно, меру условного риска можно интерпретировать как классическую выпуклую меру риска в подходящей теоретико-множественной модели. Как следствие, многие свойства меры условного риска могут быть получены путем интерпретации свойств выпуклой меры риска. Иными словами, интерпретация теоремы о двойственном представлении выпуклой меры риска приводит к новой теореме о двойственном представлении меры условного риска. В качестве примера приложения устанавливается общую теорема устойчивости представлении меры условного риска и изучаются различные ее частные случаи.
Ключевые слова: булевозначный анализ, мера условного риска, теория двойственности, принцип переноса.
Образец цитирования: Zapata J. M. A Boolean Valued Analysis Approach to Conditional Risk // Владикавк. мат. журн. 2019. Т. 21, № 4. C. 71-89 (in English). DOI 10.23671/VNC.2019.21.44629
1. Artzner, P., Delbaen, F., Eber, J. M. and Heath, D.
Coherent Measures of Risk, Mathematical Finance,
1999, vol. 9, no. 3, pp. 203-228. DOI: 10.1111/1467-9965.00068.
2. Arai, T. Convex Risk Measures on Orlicz Spaces: Inf-Convolution and Shortfall,
Mathematics and Financial Economics, 2010, vol. 3, no. 2, pp. 73-88.
DOI: 10.1007/s11579-010-0028-8.
3. Arai, T. and Fukasawa, M. Convex Risk Measures for Good Deal Bounds,
Mathematical Finance, 2014, vol. 24, no. 3, pp. 464-484.
DOI: 10.1111/mafi.12020.
4. Biagini, S. and Frittelli, M. On the extension of the Namioka-Klee Theorem and on the Fatou
Property for Risk Measures, Optimality and Risk-Modern Trends in Mathematical Finance,
2009, pp. 1-28. DOI: 10.1007/978-3-642-02608-9_1.
5. Cheridito, P. and Li, T. Dual Characterization of Properties of Risk Measures on Orlicz
Hearts, Mathematics and Financial Economics,
2008, vol. 2, no. 1, pp. 29-55. DOI: 10.1007/s11579-008-0013-7.
6. Cheridito, P. and Li, T. Risk Measures on Orlicz Hearts,
Mathematical Finance, 2009, vol. 19, no. 2, pp. 189-214.
DOI: 10.1111/j.1467-9965.2009.00364.x.
7. Delbaen, F. Risk Measures for Non-Integrable Random Variables,
Mathematical Finance, 2009, vol. 19, no. 2, pp. 329-333.
DOI: 10.1111/j.1467-9965.2009.00370.x.
8. Farkas, W., Koch-Medina, P. and Munari, C. Beyond Cash-Additive Risk Measures:
When Changing the Nume raire Fails, Finance and Stochastics,
2014, vol. 18, no. 1, pp. 145-173. DOI: 10.1007/s00780-013-0220-9.
9. Frittelli, M. and Gianin, E. R. Putting Order in Risk Measures,
Journal of Banking & Finance, 2002, vol. 26, no. 7, pp. 1473-1486.
DOI: 10.1016/S0378-4266(02)00270-4.
10. Jouini, E., Schachermayer, W., and Touzi, N. Law Invariant Risk Measures have the Fatou Property,
Advances in Mathematical Economics, 2006, vol. 9, pp. 49-72.
DOI: 10.1007/4-431-34342-3_4.
11. Kaina, M. and Ru schendorf, L. On Convex Risk Measures on \(L^p\)-Spaces,
Mathematical Methods of Operations Research, 2009, vol. 69, no. 3, pp. 475-495.
DOI: 10.1007/s00186-008-0248-3.
12. Orihuela, J. and Rui z, M. Lebesgue Property for Convex Risk Measures on Orlicz Spaces,
Mathematics and Financial Economics, 2012, vol. 6, no. 1, pp. 15-35.
DOI: 10.1007/s11579-012-0058-5.
13. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. and Ku, H.
Coherent Multiperiod Risk Adjusted Values and Bellman's Principle,
Annals of Operations Research, 2007, vol. 152, no. 1, pp. 5-22.
DOI: 0.1007/s10479-006-0132-6.
14. Biagini, S. and Bion-Nadal, J. Dynamic Quasi Concave Performance Measures,
Journal of Mathematical Economics, 2014, vol. 55, pp. 143-153.
DOI: 10.1016/j.jmateco.2014.02.007.
15. Bielecki, T. R., Cialenco, I., Drapeau, S. and Karliczek, M.
Dynamic Assessment Indices, Stochastics, 2016, vol. 88, no. 1, pp. 1-44.
DOI: 10.1080/17442508.2015.1026346.
16. Bion-Nadal, J. Dynamic Risk Measures: Time Consistency and Risk Measures from BMO
Martingales, Finance and Stochastics, 2008, vol. 12, no. 2, pp. 219-244.
DOI: 10.1007/s00780-007-0057-1.
17. Detlefsen, K. and Scandolo, G. Conditional and Dynamic Convex Risk Measures,
Finance and Stochastics, 2005, vol. 9, no. 4, pp. 539-561.
DOI: 10.1007/s00780-005-0159-6.
18. Bion-Nadal, J. Conditional Risk Measures and Robust Representation of Convex
Conditional Risk Measures, Preprint, 2004.
19. Filipovic, D., Kupper, M. and Vogelpoth, N. Approaches to Conditional Risk,
SIAM Journal of Financial Mathematics, 2012, vol. 3, no. 1, pp. 402-432.
DOI: 10.1137/090773076.
20. Follmer, H. and Penner, I. Convex Risk Measures and the Dynamics of Their Penalty Functions,
Statistics & Decisions, 2006, vol. 24, no. 1, pp. 61-96.
DOI: 10.1524/stnd.2006.24.1.61.
21. Frittelli, M. and Maggis, M. Conditional Certainty Equivalent,
International Journal of Theoretical and Applied Finance,
2011, vol. 14, no. 1, pp. 41-59. DOI: 10.1142/S0219024911006255.
22. Frittelli, M. and Maggis, M. Complete Duality for Quasiconvex Dynamic Risk Measures on Modules of
the \(L^{p} \)-Type, Statistics & Risk Modeling, 2014, vol. 31, no. 1, pp. 103-128.
DOI: 10.1515/strm-2013-1163.
23. Filipoviс, D., Kupper, M. and Vogelpoth, N. Separation and Duality
in Locally \(L^0\)-Convex Modules, Journal of Functional Analysis,
2009, vol. 256, no. 12, pp. 3996-4029.
DOI: 10.1016/j.jfa.2008.11.015.
24. Cheridito, P., Kupper, M. and Vogelpoth, N. Conditional Analysis on \(\mathbb{R ^d}\),
Set Optimization-State of the Art and Applications in Finance,
Springer Berlin Heidelberg, 2015, pp. 179-211. DOI: 10.1007/978-3-662-48670-2_6
25. Guo, T. Relations Between Some Basic Results Derived from Two Kinds of
Topologies for a Random Locally Convex Module, Journal of Functional Analysis,
2010, vol. 258, no. 9, pp. 3024-3047.
DOI: 10.1016/j.jfa.2010.02.002.
26. Drapeau, S., Jamneshan, A., Karliczek, M. and Kupper, M.
The Algebra of Conditional Sets, and the Concepts of Conditional
Topology and Compactness, Journal of Mathematical Analysis and Applications,
2016, vol. 437, no. 1, pp. 561-589. DOI: 10.1016/j.jmaa.2015.11.057.
27. Cerreia-Vioglio, S, Kupper, M., Maccheroni, F., Marinacci, M. and Vogelpoth, N.
Conditional \(L_p\)-Spaces and the Duality of Modules Over
\(f\)-Algebras, Journal of Mathematical Analysis and Applications,
2016, vol. 444, no. 2, pp. 1045-1070.
DOI: 10.1016/j.jmaa.2016.06.018.
28. Cerreia-Vioglio, S., Maccheroni, F. and Marinacci, M.
Hilbert \(A\)-Modules, Journal of Mathematical Analysis and Applications,
2017, vol. 446, no. 1, pp. 970-1017. DOI: 10.1016/j.jmaa.2016.07.046.
29. Eisele, K-T. and Taieb, S. Weak Topologies for Modules Over Rings of Bounded Random Variables,
Journal of Mathematical Analysis and Applications,
2015, vol. 421. no. 2, pp. 1334-1357. DOI: 10.1016/j.jmaa.2014.07.062.
30. Aviles, A. and Zapata, J. M. Boolean-Valued Models as a Foundation for Locally \(L^0\)-Convex
Analysis and Conditional Set Theory, Journal of Applied Logics, 2018, vol. 5, no. 1, pp. 389-420.
31. Cohen, P. J. Set Theory and the Continuum Hypothesis,
W. A. Benjamin Inc., New York and Amsterdam, 1966.
32. Gordon, E. I. Real Numbers in Boolean Valued Models of Set Theory and \(K\)-Spaces,
Dokl. Akad. Nauk SSSR, 1977, vol. 237, no. 4, pp. 777-780.
33. Takeuti, G. Two Applications of Logic to Mathematics,
Princeton University Press, 1978.
34. Kusraev, A. G. and Kutateladze, S. S. Boolean Valued Analysis,
Dordrecht,Springer, 2012. DOI: 10.1007/978-94-011-4443-8.
35. Follmer, H. and Schied, A. Stochastic Finance. An Introduction in Discrete Time,
3rd edition, Walter de Gruyter, Berlin, New York, 2011.
36. Follmer, H. and Schied, A. Convex Measures of Risk and Trading Constraints,
Finance and Stochastics, 2002, vol. 6, no. 4, pp. 429-447.
DOI: 10.1007/s007800200072.
37. Frittelli, M. and Rosazza Gianin, E. Putting Order in Risk Measures,
Journal of Banking & Finance, 2002, vol. 26, no. 7, pp. 1473-1486.
DOI: 10.1016/S0378-4266(02)00270-4.
38. Zaanen, A. C. Introduction to Operator Theory in Riesz Spaces,
Springer Science & Business Media, 2012.
DOI: 10.1007/978-3-642-60637-3.
39. Luxemburg, W. A. J. Banach Function Spaces, PhD thesis, Delft, 1955.
40. Zaanen, A. C. Riesz Spaces, vol. II, North-Holland, Amsterdam, 1983.
DOI: 10.1016/s0924-6509(08)70245-9.
41. Owari, K. On the Lebesgue Property of Monotone Convex Functions,
Mathematics and Financial Economics, 2014, vol. 8, no. 2, pp. 159-167.
DOI: 10.1007/s11579-013-0111-z.
42. Vogelpoth, N. \(L^0\)-Convex Analysis and Conditional Risk Measures,
PhD thesis, Uniwien, 2009.
43. Kusraev, A. G. Vector Duality and its Applications,
Nauka, Novosibirsk, 1985 [in Russian].
44. Kusraev, A. G. Boolean valued Analysis of Duality Between Universally Complete
Modules, Dokl. Akad. Nauk SSSR, 1982, vol. 267, no. 5, pp 1049-1052.
45. Jamneshan, A. and Zapata, J. M. On Compactness in Topological \(L^0\)-Modules,
arXiv-preprint arXiv:2084347, 2017.
46. Delbaen, F. Differentiability Properties of Utility Functions,
Optimality and Risk-Modern Trends in Mathematical Finance,
Springer Berlin Heidelberg, 2009, pp. 39-48.
DOI: 10.1007/978-3-642-02608-9_3.
47. Zapata, J. M. A Boolean-valued Models Approach to \(L^0\)-Convex Analysis,
Conditional Risk and Stochastic Control, PhD thesis, Universidad de Murcia, 2018.
48. Takeuti, G. Two Applications of Logic to Mathematics,
Publications of the Mathematical Society of Japan, Princeton
University Press, 2015. DOI: 10.1515/9781400871346.
49. Gordon, E. I. \(K\)-Spaces in Boolean Valued Models of Set Theory,
Dokl. Akad. Nauk SSSR, 1981, vol. 258, no. 4, pp. 777-780.
50. Kusraev, A. G. and Kutateladze, S. S. Boolean Valued Analysis: Selected Topics,
Vladikavkaz, Southern Mathematical Institute Press, 2013.
