Abstract
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization—attainment of infima and a version of the Minimax theorem—are given. Moreover, a new fixed-point theorem of the Knaster-Kuratowski-Mazurkiewicz-type is derived and used to prove a general version of the Walrasian excess-demand theorem.
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References
Arrow K., Debreu G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)
Bewley T.: Existence of equilibria in economies with infinitely many commodities. J. Econ. Theory 43, 514–540 (1972)
Billingsley, P.: Probability and measure. Wiley Series in Probability and Mathematical Statistics. Wiley, A Wiley-Interscience Publication, New York, Third edition, 1995
Buhvalov A.V., Lozanovskiĭ G.Ja.: Sets closed in measure in spaces of measurable functions. Dokl. Akad. Nauk SSSR 212, 1273–1275 (1973)
Cox J.C., Huang C.F.: Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory 49, 33–83 (1989)
Cvitanić J., Schachermayer W., Wang H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 237–259 (2001)
Delbaen, F., Schachermayer, W.: A compactness principle for bounded sequences of martingales with applications. In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996), volume 45 of Progr Probab, pp 137–173. Birkhäuser, Basel (1999)
Delbaen F., Schachermayer W.: The Mathematics of Arbitrage. Springer Finance, Springer-Verlag, Berlin (2006)
Delbaen F., Schachermayer W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)
Girardi, M.: Weak vs. norm compactness in L 1: the Bocce criterion. Studia Math. 98(1): 95–97, (1991). ISSN 0039-3223
Granas, A., Dugundji, J.: Fixed point theory. Springer Monographs in Mathematics. Springer-Verlag, New York, (2003). ISBN 0-387-00173-5
He H., Pearson N.D.: Consumption and portfolio policies with incomplete markets and short-sale constraints: the infinite-dimensional case. J. Econ. Theory 54, 259–304 (1991)
Kalton, N.J., Peck, N.T., Roberts, James W.: An F-space sampler, volume 89 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1984) ISBN 0-521-27585-7
Karatzas I., Žitković G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31(4), 1821–1858 (2003)
Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.-L.: Optimality conditions for utility maximization in an incomplete market. In Analysis and optimization of systems (Antibes, 1990), volume 144 of Lecture Notes in Control and Inform. Sci., pp. 3–23. Springer, Berlin (1990)
Kardaras, C., Žitković, G.: Stability of the utility maximization problem with random endowment in incomplete markets. to appear in Mathematical Finance, 2007. arxiv:0706.0482
Knaster, B., Kuratowski, C., Mazurkiewicz, S.: Ein Beweis des Fixpunktsatsez für n-dimensionale Simplexe. Fund. Math., XIV: 132–137 (1929)
Komlós J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967)
Kramkov D., Schachermayer W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999)
Larsen K., Žitković G.: Stability of utility-maximization in incomplete markets. Stoch. Process. Appl. 117(11), 1642–1662 (2007)
Mas-Colell, A., Zame, W.R.: Equilibrium theory in infinite-dimensional spaces. In: Handbook of mathematical economics, Vol. IV, volume 1 of Handbooks in Economics, pp 1835–1898. North-Holland, Amsterdam (1991)
McKenzie L.W.: On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71 (1959)
Merton R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Statist. 51, 247–257 (1969)
Merton R.C.: Optimum consumption and portfolio rules in a conitinuous-time model. J. Econ. Theory 3, 373–413 (1971)
Nikišin E.M.: A certain problem of Banach. Dokl. Akad. Nauk SSSR 196, 774–775 (1971)
Pliska S.R.: A stochastic calculus model of continuous trading: optimal portfolio. Math. Oper. Res. 11, 371–382 (1986)
Schwartz M.: New proofs of a theorem of Komlós. Acta Math. Hung. 47, 181–185 (1986)
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Žitković, G. Convex compactness and its applications. Math Finan Econ 3, 1–12 (2010). https://doi.org/10.1007/s11579-010-0024-z
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DOI: https://doi.org/10.1007/s11579-010-0024-z