Stationarity of a GARCH Recurrence

Definitions and mathematical statement

Put $A_t:=\alpha_0 Z_t^2+\beta_0$. The supplied recurrence and logarithmic condition are \[ h_t=\omega_0+A_{t-1}h_{t-1},\qquad {\mathbb{E}}[\log A_t]<0. \] Here $\omega_0,\alpha_0,\beta_0$ are parameters and $(Z_t)$ is a stochastic sequence, but their domains, parameter restrictions, probability law, and time index set are not specified. The logarithmic condition requires a convention under which the expectation is defined; the source does not specify positivity, integrability, or treatment of $\log 0$.

Reported implication to be clarified. The record asks why this logarithmic condition implies the existence of a stationary ergodic solution and convergence of solutions from different initial conditions. For a two-sided interpretation, a solution would be a process $(H_t)_{t\in{\mathbb{Z}}}$, defined jointly with the driving sequence, satisfying \[ H_t=\omega_0+A_{t-1}H_{t-1}\quad\text{almost surely for every }t\in{\mathbb{Z}}. \] Strict stationarity means that every finite-dimensional distribution is unchanged by integer shifts of all its time indices. Ergodicity means that each shift-invariant event under the process law has probability zero or one. The source does not specify whether these properties concern $(H_t)$ alone or its joint process with $(Z_t)$, nor whether a two-sided time index is intended.

A possible interpretation of convergence from different initial conditions is the following: for each pair of admissible initial values $x,y$, solutions $(h_t^{(x)})$ and $(h_t^{(y)})$ driven by the same sequence satisfy \[ \lvert h_t^{(x)}-h_t^{(y)}\rvert\longrightarrow0\qquad(t\to\infty). \] This interpretation is provisional: admissible initial values, their dependence on the driving sequence, the coupling of solutions, and the mode of convergence are absent from the supplied formulation.

Unresolved qualifications. No independence, identical-distribution, stationarity, or ergodicity assumptions on $(Z_t)$ are supplied. Consequently, the displayed logarithmic condition and recurrence do not specify a complete implication with justified hypotheses. This draft records the intended explanatory question without asserting it as a theorem or as a verified open problem.

Short English statement

The record asks why a negative expected logarithmic multiplier yields a stationary ergodic solution of the stated recurrence and convergence from different initial conditions. Essential assumptions on the noise, parameters, logarithm, and convergence are missing. Supplied verification describes this as a standard explanatory step; the registry's open label is not an independently verified mathematical status.

Sources

• Local problem registry, permanent UnsolvedMath ID $30002414$, title Stationarity of a GARCH Recurrence, exact source code catalogue entry; supplied collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2013.
• Supplied report citation: Statistical Inference for Complex Time Series Data (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/48, pp. 2749--2823. https://doi.org/10.4171/owr/2013/48.
• Supplied literature source, labeled "Oberwolfach Report": https://ems.press/content/serial-article-files/46479.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the record open and says that only the fragment "Why does this hold" was available and its antecedent was omitted. This conflicts with the explicit recurrence present in the supplied statement.
• Supplied statement-verification note reports checking the official Oberwolfach report and describes the implication as a standard explanatory step establishing stationarity and initial-condition stability, rather than an unresolved problem. It says that no separate clean statement was promoted and that the original record remains visible for provenance. This draft has not independently visited or verified either external source.