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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Stochastic equilibria in von Neumann–Gale dynamical systems
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by Igor V. Evstigneev and Klaus Reiner Schenk-Hoppé PDF
Trans. Amer. Math. Soc. 360 (2008), 3345-3364 Request permission


This paper examines a class of random dynamical systems related to the classical von Neumann and Gale models of economic dynamics. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. We establish a general existence theorem for equilibrium, which holds under conditions analogous to the standard deterministic ones. Our results answer questions that remained open for more than three decades.
  • V. I. Arkin and I. V. Evstigneev. Stochastic models of control and economic dynamics. Academic Press, 1987.
  • L. Arnold, I. V. Evstigneev, and V. M. Gundlach, Convex-valued random dynamical systems: a variational principle for equilibrium states, Random Oper. Stochastic Equations 7 (1999), no. 1, 23–38. MR 1677758, DOI 10.1515/rose.1999.7.1.23
  • Ludwig Arnold, Volker Matthias Gundlach, and Lloyd Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab. 4 (1994), no. 3, 859–901. MR 1284989
  • M. A. H. Dempster, I. V. Evstigneev and M. I. Taksar. Asset pricing and hedging in financial markets with transaction costs: An approach based on the von Neumann-Gale model. Annals of Finance 2 (2006), 327–355.
  • A. Dvoretzky, A. Wald, and J. Wolfowitz, Elimination of randomization in certain problems of statistics and of the theory of games, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 256–260. MR 35979, DOI 10.1073/pnas.36.4.256
  • E. B. Dynkin. Some probability models for a developing economy. Soviet Mathematics Doklady 12 (1971), 1422–1425.
  • E. B. Dynkin. Stochastic concave dynamic programming. Math. USSR Sbornik 16 (1972), 501-515.
  • E. B. Dynkin and A. A. Yushkevich, Controlled Markov processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 235, Springer-Verlag, Berlin-New York, 1979. Translated from the Russian original by J. M. Danskin and C. Holland. MR 554083
  • I. V. Evstigneev, Positive matrix-valued cocycles over dynamical systems, Uspehi Mat. Nauk 29 (1974), no. 5(179), 219–220 (Russian). MR 0396906
  • I. V. Evstigneev and S. D. Flåm. Stochastic programming: Non-anticipativity and Lagrange multipliers. In: Encyclopedia of Optimization, Kluwer Academic Publishers, Vol. 4, 2001, pp. 332-338.
  • I. V. Evstigneev and K. R. Schenk-Hoppé. The von Neumann-Gale growth model and its stochastic generalization. In: R. Dana, C. Le Van, T. Mitra, K. Nishimura (eds.), Handbook on Optimal Growth, Vol. I, Chapter 2, Springer, 2006, pp. 337-383.
  • Igor V. Evstigneev and Klaus Reiner Schenk-Hoppé, Pure and randomized equilibria in the stochastic von Neumann-Gale model, J. Math. Econom. 43 (2007), no. 7-8, 871–887. MR 2341683, DOI 10.1016/j.jmateco.2007.04.004
  • Igor V. Evstigneev and Michael I. Taksar, Rapid growth paths in convex-valued random dynamical systems, Stoch. Dyn. 1 (2001), no. 4, 493–509. MR 1875064, DOI 10.1142/S0219493701000242
  • David Gale, The closed linear model of production, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, pp. 285–303. MR 0085961
  • David Gale, A mathematical theory of optimal economic development, Bull. Amer. Math. Soc. 74 (1968), 207–223. MR 221835, DOI 10.1090/S0002-9904-1968-11891-9
  • D. Gale. A note on the nonexistence of optimal price vectors in the general balanced-growth model of Gale: Comment. Econometrica 40 (1972), 391–392.
  • J. Hülsmann and V. Steinmetz, A note on the nonexistence of optimal price vectors in the general balanced-growth model of Gale, Econometrica 40 (1972), 387–389. MR 378726, DOI 10.2307/1909414
  • Yuri Kifer, Perron-Frobenius theorem, large deviations, and random perturbations in random environments, Math. Z. 222 (1996), no. 4, 677–698. MR 1406273, DOI 10.1007/PL00004551
  • J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229. MR 210177, DOI 10.1007/BF02020976
  • Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
  • V. L. Makarov and A. M. Rubinov, Mathematical theory of economic dynamics and equilibria, Springer-Verlag, New York-Heidelberg, 1977. Translated from the Russian by Mohamed El-Hodiri. MR 0439072
  • Hukukane Nikaidô, Convex structures and economic theory, Mathematics in Science and Engineering, Vol. 51, Academic Press, New York-London, 1968. MR 0277233
  • Roy Radner, Balanced stochastic growth at the maximum rate, Contributions to the von Neumann growth model (Proc. Conf., Inst. Advanced Studies, Vienna, 1970) Zeitschrift für Nationalökonomie, Supplementum 1, Springer, New York, 1971, pp. 39–52. MR 0325117
  • Roy Radner, Optimal steady-state behavior of an economy with stochastic production and resources, Mathematical topics in economic theory and computation (Sympos. Math. Econom., SIAM Fall Meeting, Univ. Wisconsin, Madison, Wis., 1971) Soc. Indust. Appl. Math., Philadelphia, Pa., 1972, pp. 99–112. MR 0406415
  • Roy Radner, Optimal stationary consumption with stochastic production and resources, J. Econom. Theory 6 (1973), no. 1, 68–90. MR 452560, DOI 10.1016/0022-0531(73)90043-4
  • R. Radner. Equilibrium under uncertainty. In: K. J. Arrow and M. D. Intrilligator (eds.), Handbook of Mathematical Economics, North-Holland, 1982, pp. 923–1006.
  • R. Tyrrell Rockafellar, Monotone processes of convex and concave type, Memoirs of the American Mathematical Society, No. 77, American Mathematical Society, Providence, R.I., 1967. MR 0225231
  • R. T. Rockafellar and R. J.-B. Wets, Nonanticipativity and ${\cal L}^{1}$-martingales in stochastic optimization problems, Math. Programming Stud. 6 (1976), 170–187. MR 462590
  • J. von Neumann. Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In: Ergebnisse eines Mathematischen Kolloquiums, No. 8, 1935-1936, Franz-Deuticke, 1937, pp. 73–83. [Translated: A model of general economic equilibrium, Review of Economic Studies 13 (1945-1946), 1–9.]
  • Kôsaku Yosida and Edwin Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66. MR 45194, DOI 10.1090/S0002-9947-1952-0045194-X
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Additional Information
  • Igor V. Evstigneev
  • Affiliation: Economics Department, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
  • MR Author ID: 210292
  • Email:
  • Klaus Reiner Schenk-Hoppé
  • Affiliation: School of Mathematics and Leeds University Business School, Leeds University, Leeds LS2 9JT, United Kingdom
  • Email:
  • Received by editor(s): July 5, 2006
  • Received by editor(s) in revised form: October 27, 2006
  • Published electronically: January 11, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 3345-3364
  • MSC (2000): Primary 37H99, 37H15; Secondary 91B62, 91B28
  • DOI:
  • MathSciNet review: 2379800