The von Neumann-Gale Growth Model and Its Stochastic Generalization

  • Igor V. Evstigneev
  • Klaus R. Schenk-Hoppé


Income Expense Nash Volatility Librium 
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  1. [1]
    Aghion, P., and P. Howitt: Endogenous Growth Theory, MIT Press, Cambridge, 1998.Google Scholar
  2. [2]
    Akin, E.: The General Topology of Dynamical Systems, American Mathematical Society, Providence, 1993.Google Scholar
  3. [3]
    Algoet, P.H., and T.M. Cover: “Asymptotic optimality and asymptotic equipartition properties of log-optimum investment,” Annals of Probability 16 (1988), 876–898.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Amir, R.: “Sensitivity analysis in multisector optimal economic dynamics,” Journal of Mathematical Economics 25 (1996), 123–141.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Amir, R., and I.V. Evstigneev: “A functional central limit theorem for equilibrium paths of economic dynamics,” Journal of Mathematical Economics 33 (2000), 81–99.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Anoulova, S.V., I.V. Evstigneev, and V.M. Gundlach: “Turnpike theorems for positive multivalued stochastic operators,” Advances in Mathematical Economics 2 (2000), 1–20.MathSciNetGoogle Scholar
  7. [7]
    Arkin, V.I., and I.V. Evstigneev: Stochastic Models of Control and Economic Dynamics, Academic Press, London, 1987.Google Scholar
  8. [8]
    Arnold, L.: Random Dynamical Systems, Springer-Verlag, Berlin, 1998.Google Scholar
  9. [9]
    Arnold, L., I.V. Evstigneev, and V.M. Gundlach: “Convex-valued random dynamical systems: A variational principle for equilibrium states,” Random Operators and Stochastic Equations 7 (1999), 23–38.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Arnold, L., V.M. Gundlach, and L. Demetrius: “Evolutionary formalism for products of positive random matrices,” Annals of Applied Probability 4 (1994), 859–901.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Barro, R.J., and X. Sala-i-Martin: Economic Growth, McGraw-Hill, New York, 1995.Google Scholar
  12. [12]
    Belenky, V.Z.: “A stochastic stationary model for optimal control of an economy,” in Studies in Stochastic Control Theory and Mathematical Economics (N.Ya. Petrakov et al., eds.), pages 3–24, CEMI, Moscow, 1981 (in Russian).Google Scholar
  13. [13]
    Brock, W.A., and W.D. Dechert: Growth Theory, Nonlinear Dynamics and Economic Modelling: Scientific Essays of William Allen Brock (Economists of the Twentieth Century), Edward Elgar Publ., Cheltenham, 2001.Google Scholar
  14. [14]
    de Hek, P.: “On endogenous growth under uncertainty,” International Economic Review 40 (1999), 727–744.CrossRefADSGoogle Scholar
  15. [15]
    de Hek, P., and S. Roy: “On sustained growth under uncertainty,” International Economic Review 42 (2001), 801–814.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Dempster, M.A.H., I.V. Evstigneev, and K.R. Schenk-Hoppé: “Exponential growth of fixed-mix strategies in stationary asset markets,” Finance and Stochastics 7 (2003), 263–276.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Dempster, M.A.H., I.V. Evstigneev, and K.R. Schenk-Hoppé: “Volatility-induced financial growth,” Working Paper No. 10/2004, Institute for Financial Research, University of Cambridge, 2004.Google Scholar
  18. [18]
    Dynkin, E.B.: “Some probability models for a developing economy,” Soviet Mathematics Doklady 12 (1971), 1422–1425.zbMATHGoogle Scholar
  19. [19]
    Dynkin, E.B., and A.A. Yushkevich: Controlled Markov Processes and Their Applications, Springer-Verlag, New York, 1979.Google Scholar
  20. [20]
    Evstigneev, I.V.: “Positive matrix-valued cocycles over dynamical systems,” Uspekhi Matematicheskikh Nauk 29 (1974), 219–220 (in Russian).zbMATHMathSciNetGoogle Scholar
  21. [21]
    Evstigneev, I.V.: “Homogeneous convex models in the theory of controlled random processes,” Soviet Mathematics Doklady 22 (1980), 108–111.zbMATHGoogle Scholar
  22. [22]
    Evstigneev, I.V., and S.D. Flåm: “Rapid growth paths in multivalued dynamical systems generated by homogeneous convex stochastic operators,” Set-Valued Analysis 6 (1998), 61–82.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Evstigneev, I.V., and Yu.M. Kabanov: “Probabilistic modification of the von Neumann-Gale model,” Russian Mathematical Surveys 35 (1980), 185–186.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Evstigneev, I.V., and S.E. Kuznetsov: “Probabilistic variant of the turnpike theorem for homogeneous convex controllable models,” Mathematical Notes 33 (1983), 185–194.MathSciNetGoogle Scholar
  25. [25]
    Evstigneev, I.V., and K.R. Schenk-Hoppé: “From rags to riches: On constant proportions investment strategies,” International Journal of Theoretical and Applied Finance 5 (2002), 563–573.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Evstigneev, I.V., and K.R. Schenk-Hoppé: “Pure and randomized equilibria in the stochastic von Neumann-Gale model,” Discussion Paper 0507, School of Economic Studies, University of Manchester, 2005.Google Scholar
  27. [27]
    Evstigneev, I.V., and M.I. Taksar: “Rapid growth paths in convex-valued random dynamical systems,” Stochastics and Dynamics 1 (2001), 493–509.MathSciNetCrossRefGoogle Scholar
  28. [28]
    Evstigneev, I.V., and M.I. Taksar: “Asset pricing and hedging under transaction costs: An approach based on the von Neumann-Gale model,” Discussion Paper 0422, School of Economic Studies, University of Manchester, 2004.Google Scholar
  29. [29]
    Föllmer, H., and A. Schied: Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter, Berlin, 2002.CrossRefGoogle Scholar
  30. [30]
    Gale, D.: “A closed linear model of production,” in: Linear Inequalities and Related Systems (H.W. Kuhn and A.W. Tucker, eds.), pages 285–303, Princeton University Press, Princeton, 1956.Google Scholar
  31. [31]
    Gale, D.: “A mathematical theory of optimal economic development,” Bulletin of the American Mathematical Society 74 (1968), 207–223.zbMATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    Gale, D.: “A note on the nonexistence of optimal price vectors in the general balanced-growth model of Gale: Comment,” Econometrica 40 (1972), 391–392.MathSciNetCrossRefGoogle Scholar
  33. [33]
    Hakansson, N.H., and W.T. Ziemba: “Capital growth theory,” in: Handbooks in Operations Research and Management Science, Volume 9, Finance (R.A. Jarrow, V. Maksimovic, W.T. Ziemba, eds.), Chapter 3, pages 65–86, Elsevier, Amsterdam, 1995.Google Scholar
  34. [34]
    Hulsmann, J., 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.MathSciNetCrossRefGoogle Scholar
  35. [35]
    Iyengar, G., and T.M. Cover: “Growth optimal investment in horse race markets with costs,” IEEE Transactions on Information Theory 46 (2000), 2675–2683.MathSciNetCrossRefGoogle Scholar
  36. [36]
    Kabanov, Yu.M.: “Hedging and liquidation under transaction costs in currency markets,” Finance and Stochastics 3 (1999), 237–248.zbMATHCrossRefGoogle Scholar
  37. [37]
    Kabanov, Yu.M.: “The arbitrage theory,” in: Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management (E. Jouini, J. Cvitanić and M. Musiela, eds.), pages 3–42, Cambridge University Press, Cambridge, 2001.Google Scholar
  38. [38]
    Kabanov, Yu.M. and C. Stricker: “The Harrison-Pliska arbitrage pricing theorem under transaction costs,” Journal of Mathematical Economics 35 (2001), 185–196.MathSciNetCrossRefGoogle Scholar
  39. [39]
    Long, J.B.: “The numeraire portfolio,” Journal of Financial Economics 26 (1990), 29–69.CrossRefGoogle Scholar
  40. [40]
    Luenberger, D.G.: Optimization by Vector Space Methods, Wiley, New York, 1969.Google Scholar
  41. [41]
    Makarov, V.L., and A.M. Rubinov: Mathematical Theory of Economic Dynamics and Equilibria, Springer-Verlag, Berlin, 1977.Google Scholar
  42. [42]
    McKenzie, L.W.: “Optimal economic growth, turnpike theorems and comparative dynamics,” in Handbook of Mathematical Economics: Volume III (K.J. Arrow and M.D. Intriligator, eds.), pages 1281–1355, North-Holland, Amsterdam, 1986.Google Scholar
  43. [43]
    McKenzie, L.W.: “Turnpikes,” American Economic Review Papers and Proceedings 88 (1998), 1–14.Google Scholar
  44. [44]
    Mirman, L.J.: “One sector economic growth and uncertainty: a survey,” in Stochastic Programming (M.A.H. Dempster, ed.), pages 537–567, Academic Press, London, 1980.Google Scholar
  45. [45]
    Mitra, T., L. Montrucchio, and F. Privileggi: “The nature of steady states in models of optimal growth under uncertainty,” Economic Theory 23 (2003), 39–71.MathSciNetCrossRefGoogle Scholar
  46. [46]
    Neveu, J.: Mathematical Foundations of the Calculus of Probability Theory, Holden Day, San Francisco, 1965.Google Scholar
  47. [47]
    Nikaido, H.: Convex Structures and Economic Theory, Academic Press, London, 1968.Google Scholar
  48. [48]
    Nussbaum, R.D., and S.M. Verduyn Lunel: Generalizations of the Perron-Frobenius Theorem for Nonlinear Maps, Memoirs of the American Mathematical Society, Volume 138, American Mathematical Society, Providence, 1999.Google Scholar
  49. [49]
    Olson, L.J., and S. Roy: “Theory of stochastic optimal economic growth,” this volume 2005.Google Scholar
  50. [50]
    Presman, E.L., and A.D. Slastnikov: “Growth rates and optimal paths in stochastic models of expanding economy,” in Proceedings of the International Conference “Stochastic Optimization,” Kiev, 1984 (V.I. Arkin, A. Shiraev and R. Wets, eds.), Lecture Notes in Control and Information Sciences, pages 327–332, Spinger-Verlag, Berlin, 1986.Google Scholar
  51. [51]
    Radner, R.: “Balanced stochastic growth at the maximum rate,” in: Contributions to the von Neumann Growth Model (Proc. Conf., Inst. Adv. Studies, Vienna, 1970), Zeitschrift für Nationalökonomie Suppl. No. 1 (1971), 39–53.Google Scholar
  52. [52]
    Radner, R.: “Optimal steady-state behaviour of an economy with stochastic production and resources,” in: Mathematical Topics in Economic Theory and Computation (R.H. Day and S.M. Robinson, eds.), pages 99–112, SIAM, Philadelphia, 1972.Google Scholar
  53. [53]
    Ramsey, F.: “A mathematical theory of savings,” Economic Journal 38 (1928), 543–559.CrossRefGoogle Scholar
  54. [54]
    Rockafellar, R.T.: Monotone Processes of Convex and Concave Type, Memoirs of the American Mathematical Society, Volume 77, American Mathematical Society, Providence, 1967.Google Scholar
  55. [55]
    Schachermayer, W.: “The Fundamental Theorem of Asset Pricing under proportional transaction costs in finite discrete time,” Mathematical Finance 14 (2004), 19–48.zbMATHMathSciNetCrossRefGoogle Scholar
  56. [56]
    Solow, R.M.: “A contribution to the theory of economic growth,” Quarterly Journal of Economics 70 (1956), 65–94.CrossRefGoogle Scholar
  57. [57]
    Solow, R.M., and P.A. Samuelson: “Balanced growth under constant returns to scale,” Econometrica 21 (1953), 412–424.MathSciNetCrossRefGoogle Scholar
  58. [58]
    Stachurski, J.: “Stochastic growth: asymptotic distributions,” Economic Theory 21 (2003), 913–919.zbMATHMathSciNetCrossRefGoogle Scholar
  59. [59]
    Stokey, N.L., R.E. Lucas, and E.C. Prescott: Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge, 1989.Google Scholar
  60. [60]
    von Neumann, J.: “Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes,” in Ergebnisse eines Mathematischen Kolloquiums, No. 8, 1935–1936 (K. Menger, ed.), pages 73–83, Vienna: Franz-Deuticke, 1937 (in German). (Translated into English by C. Morgenstern: “A model of general economic equilibrium,” Review of Economic Studies 13 (1945–1946), 1–9.)Google Scholar

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© Springer Berlin · Heidelberg 2006

Authors and Affiliations

  • Igor V. Evstigneev
    • 1
  • Klaus R. Schenk-Hoppé
    • 2
  1. 1.Economic Studies, School of Social SciencesUniversity of ManchesterUK
  2. 2.Business School and School of MathematicsUniversity of LeedsUK

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