Summary
The existence and uniqueness of solutions of a special type of recursive stochastic equations is investigated. Such equations occur in many stochastic models in which the stochastic process describing the behaviour of the system will be generated by the so-called input.
For example, let the input be a stationary sequence {X t } t=−∞ t of random variables with values in a measurable space M and let P be the distribution of this sequence. Consider an equation of the form Zt+1 =f *(Xt,Zt). Assume that there is a system \(\{ A(t,\{ x_i \} _{i = - \infty }^{ + \infty } :t \in \Gamma , (\{ x_t \} _{t = - \infty }^{ + \infty } \in M^\Gamma \) of subsets of the state space Z with the property
for all t ε Γ,
Then, under some regularity conditions, there is a stationary solution of the given equation, i.e. there is a stationary sequence {Zt} +∞t=−∞ for which the equation almost surely holds with respect to the common distribution of ({Xt} +∞t=−∞ , {Zt} +∞t=−∞ ). Analogous results are obtained in more general models.
Article PDF
Similar content being viewed by others
References
Borovkov, A.A.: Ergodic and Stability Theorems for One Class of Stochastic Equations and Their Applications. (In Russian). Teor. Verojatnost. i Primenen. XXIII, 241–262 (1978)
Borovkov, A.A.: Continuity Theorems for Multi-server Systems without Delay. (In Russian). Teor. Verojatnost. i Primenen. XVII, 458–468 (1972)
Franken, P., Kalähne, U.: Existence, Uniqueness and Continuity of Stationary Distributions for Queueing Systems without Delay. Math. Nachr. 86, 97–115 (1978)
Franken, P., König, D., Arndt, U., Schmidt, V.: Queues and Point Processes. Akademie-Verlag, Berlin, and J. Wiley, New York 1981
Franken, P., Lisek, B.: On Wald's Identity for Dependent Variables. Z. Wahrscheinlichkeitsthcorie verw. Gebiete 60, 143–150 (1982)
Lisek, B.: Construction of Stationary State Distributions for Loss Systems. Math. Operationsforsch. Statist., Ser. Statistics, 10, no. 4, 561–581 (1979)
Lisek, B., Lisek, M.: On Qualitative Properties of Inventory Models with Discrete Demand. Elektron. Informationsverarb. Kybernetik 16, 425–432 (1980)
Loynes, R.M.: The Stability of a Queue with Non-independent Inter-arrival and Service Times. Proc. Cambridge Philos. Soc. 58, 497–520 (1962)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lisek, B. A method for solving a class of recursive stochastic equations. Z. Wahrscheinlichkeitstheorie verw Gebiete 60, 151–161 (1982). https://doi.org/10.1007/BF00531819
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00531819