Abstract
The paper examines dynamical systems generated by convex homogeneous multivalued operators in spaces of random vectors. The primary goal is to investigate the growth rates of random trajectories of these dynamical systems. Existence and characterization theorems for ‘rapid’ trajectories, growing faster in a certain sense than others, are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akin, A.: The General Topology of Dynamical Systems, Grad. Stud. Math. Vol. 1, Amer. Math. Soc., Providence, RI, 1993.
Antipin, A. S. and Flåm, S. D.: Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997), 29–41.
Arkin, V. I. and Evstigneev, I. V.: Stochastic Models of Control and Economic Dynamics, Academic Press, London, 1987.
Arnold, L., Demetrius, L. and Gundlach, M.: Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab. 4 (1994), 859–901.
Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, 1990.
Belenky, V. Z.: A stochastic stationary model for optimal control of an economy, in: N. Ya. Petrakov et al. (eds), Studies in Stochastic Control Theory and Mathematical Economics, CEMI, Moscow, 1981, pp. 3–24 (in Russian).
Castaing, C. and Valadier, M.: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.
Dubovitskii, A. Ya. and Milyutin, A. A.: Necessary conditions for a weak extremum in optimal control problems with mixed inequality constraints, Comput. Math. Math. Phys. 8 (1968), 24–98.
Dynkin, E. B.: Some probability models for a developing economy, Soviet Math. Dokl. 12 (1971), 1422–1425.
Evstigneev, I. V.: Optimal economic planning taking account of stationary random factors, Soviet Math. Dokl. 13 (1972), 1357–1359.
Evstigneev, I. V.: Positive matrix-valued cocycles over dynamical systems, Uspekhi Mat. Nauk 29 (1974), 219–220 (in Russian).
Evstigneev, I. V.: Homogeneous convex models in the theory of controlled random processes, Soviet Math. Dokl. 22 (1980), 108–111.
Evstigneev, I. V. and Kabanov, Yu. M.: Probabilistic modification of the von Neumann—Gale model, Russian Math. Surveys 35 (1980), 185–186.
Evstigneev, I. V.: A non-stationary stochastic analog of the von Neumann—Gale model of a developing economy: A theorem on a characteristic, Optimizatsiya 27 (1981), 34–43 (in Russian).
Flåm, S. D.: Nonanticipativity in stochastic programming, J. Optim. Theory Appl. 46 (1985), 23–30.
Furstenberg, H. and Kesten, H.: Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469.
Gale, D.: A closed linear model of production, in: H. W. Kuhn et al. (eds), Linear Inequalities and Related Systems, Ann. of Math. Stud. 38, Princeton Univ. Press, Princeton, 1956, pp. 285–303.
Gale, D.: A mathematical theory of optimal economic development, Bull. Amer. Math. Soc. 74 (1968), 207–223.
Hurwicz, L.: Programming in linear spaces, in: K. J. Arrow et al. (eds), Studies in Linear and Nonlinear Programming, Stanford University Press, Stanford, 1958, pp. 38–102.
Kesten, H. and Stigum, B. P.: Balanced growth under uncertainty in decomposable economies, in: M. S. Balch et al. (eds), Essays in Economic Behavior under Uncertainty, North-Holland, N.Y., 1974, pp. 339–381.
Levin, V. L.: Convex Analysis in Spaces of Measurable Functions, Nauka, Moscow, 1985 (in Russian).
Levin, V. L.: A superlinear multifunction arising in connection with mass transfer problems, Set-Valued Anal. 4 (1996), 41–65.
Luo, Z. Q., Pang, J.-S. and Ralph, D.: Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1996.
Makarov, V. L. and Rubinov, A. M.: Mathematical Theory of Economic Dynamics and Equilibria, Springer-Verlag, New York, 1977.
Makarov, V. L., Rubinov, A. M. and Levin, M. I.: Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms, Adv. Textbooks Econom. 33, Elsevier, Amsterdam, New York, 1995.
Neveu, J.: Mathematical Foundations of the Calculus of Probability Theory, Holden Day, San Francisco, 1965.
Nikaido, H.: Convex Structures and Economic Theory, Academic Press, New York, 1968.
Presman, E. L. and Slastnikov, A. D.: An approach to the definition of a growth rate in stochastic von Neumann—Gale models, in: V. I. Arkin et al. (eds), Models and Methods of Stochastic Optimization, CEMI, Moscow, 1983, pp. 123–152 (in Russian).
Radner, R.: Balanced stochastic growth at the maximum rate, in: Contributions to the von Neumann Growth Model (Proc. Conf., Inst. Adv. Studies, Vienna, 1970), Z. Nationalökonomie, Suppl. No. 1, 1971, pp. 39–53.
Radner, R.: Optimal stationary consumption with stochastic production and resources, J. Econom. Theory 6 (1973), 68–90.
Rockafellar, R. T.: Monotone Processes of Convex and Concave Type, Mem. Amer. Math. Soc. 77, Amer. Math. Soc., Providence, RI, 1967.
Rockafellar, R. T.: Integrals which are convex functionals, Pacific J. Math. 24 (1968), 525–539.
Rockafellar, R. T. and Wets, R. J.-B.: Stochastic convex programming: Kuhn-Tucker conditions, J. Math. Econom. 2 (1975), 349–370.
von Neumann, J.: Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, in: Ergebnisse eines Mathematischen Kolloquiums 8, 1935–1936, Franz-Deuticke, Leipzig und Wien, 1937, pp. 73–83. (An English translation: A model of general economic equilibrium, Rev. Econom. Studies 13 (1945–1946), 1–9.)
Yosida, K. and Hewitt, E.: Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66.
Zaharevič, M. I.: Ergodic properties of nonlinear mappings connected with models of economic dynamics, Soviet. Math. Dokl. 24 (1981), 430–433.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Evstigneev, I.V., Flå m, S.D. Rapid Growth Paths in Multivalued Dynamical Systems Generated by Homogeneous Convex Stochastic Operators. Set-Valued Analysis 6, 61–81 (1998). https://doi.org/10.1023/A:1008606332037
Issue Date:
DOI: https://doi.org/10.1023/A:1008606332037