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Stability of linear stochastic difference equations in strategically controlled random environments

Part of: Stochastic systems and control Game theory Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Ulrich Horst
Ulrich Horst*
Affiliation:
Humboldt-Universität zu Berlin
*
∗ Postal address: Institut für Mathematik, Bereich Stochastik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany. Email address: horst@mathematik.hu-berlin.de
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Abstract

We consider the stochastic sequence {Yt}t∈ℕ defined recursively by the linear relation Yt+1=AtYt+Bt in a random environment. The environment is described by the stochastic process {(At,Bt)}t∈ℕ and is under the simultaneous control of several agents playing a discounted stochastic game. We formulate sufficient conditions on the game which ensure the existence of Nash equilibria in Markov strategies which have the additional property that, in equilibrium, the process {Yt}t∈ℕ converges in distribution to a stationary regime.

Keywords

Stochastic difference equationstochastic stabilitystochastic gamesrandom systems with complete connections

MSC classification

Primary: 60G35: Signal detection and filtering 93E15: Stochastic stability 91A15: Stochastic games
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 961 - 981
Copyright
Copyright © Applied Probability Trust 2003 

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References

[1] Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S. (1988). Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré Prob. Statist. 24, 367–394.Google Scholar
[2] Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, New York.Google Scholar
[3] Brandt, A. (1986). The stochastic equation Y_n+1=A_nY_n + B_n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
[4] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
[5] Curtat, L. (1996). Markov equilibria in stochastic games with complementaries. Games Econom. Behavior 17, 177–199.Google Scholar
[6] Duffie, D., Geanakopolos, J., Mas-Colell, A. and McLennan, A. (1994). Stationary Markov equilibria. Econometrica 62, 745–781.Google Scholar
[7] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterisation and Convergence. John Wiley, New York.Google Scholar
[8] Föllmer, H. and Schweizer, M. (1993). A microeconomic approach to diffusion models for stock prices. Math. Finance 3, 1–23.Google Scholar
[9] Föllmer, H., Horst, U. and Kirman, A. (2003). Financial price fluctuation in a simple learning model. Working paper, Humboldt-Universität zu Berlin.Google Scholar
[10] Horst, U. (2001). The stochastic equation Y_t+1=A_tY_t + B_t with non-stationary coefficients. J. Appl. Prob. 38, 80–94.Google Scholar
[11] Horst, U. and Scheinkman, J. (2002). Equilibria in systems of social interactions. Working paper, Princeton University.Google Scholar
[12] Liggett, T. (1985). Interacting Particle Systems. Springer, Berlin.Google Scholar
[13] Montrucchio, L. (1987). Lipschitz continuous policy functions for strongly concave optimization problems. J. Math. Econom. 16, 259–273.Google Scholar
[14] Norman, F. M. (1972). Markov Processes and Learning Models. Academic Press, New York.Google Scholar
[15] Nowak, A. (1985). Existence of equilibrium stationary strategies in discounted non-cooperative stochastic games with uncountable state space. J. Optimization Theory Appl. 45, 591–602.Google Scholar
[16] Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar