Abstract
AbstractWe establish the existence of a unique stationary and ergodic solution for systems of stochastic recurrence equations defined by stochastic self-maps on Polish metric spaces based on the fixed point theorem of Matkowski. The results can be useful in cases where the stochastic Lipschitz coefficients implied by the currently used method either do not exist, or lead to the imposition of unnecessarily strong conditions for the derivation of the solution.
Highlights
The present note concerns the establishment of the existence of a unique stationary and ergodic solution over the set of integral numbers for systems of stochastic recurrence equations defined by stochastic self-maps on Polish metric spaces and its representation as a limit of relevant Picard iterates
The relevant supremum metric is denoted by dΘ. →eas denotes exponentially almost sure convergence, →Pa.s. denotes P a.s. convergence, and =as almost sure equality w.r.t
The following theorem establishes the existence of a unique, up to indistinguishability, stationary and ergodic solution to the stochastic recurrence system defined by xt+1 = Φt,θ, its continuity properties w.r.t. θ, the form by which it approximates any other solution as well as the issue of its invertibility
Summary
The present note concerns the establishment of the existence of a unique stationary and ergodic solution over the set of integral numbers for systems of stochastic recurrence equations defined by stochastic self-maps on Polish metric spaces and its representation as a limit of relevant Picard iterates. 2 Existence and Uniqueness of Stationary and Ergodic Solution to SRE’s The following theorem establishes the existence of a unique, up to indistinguishability, stationary and ergodic solution to the stochastic recurrence system defined by xt+1 = Φt,θ (xt), its continuity properties w.r.t. θ, the form by which it approximates any other solution as well as the issue of its invertibility.
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