Stationarity, moment properties, and particle-EM estimation for periodic logGARCH-SV models

Omar Alzeley; Omar Alzeley

doi:10.3934/math.20261266

AIMS Mathematics

2026, Volume 11, Issue 9: 32199-32222. doi: 10.3934/math.20261266

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Research article

Stationarity, moment properties, and particle-EM estimation for periodic logGARCH-SV models

Omar Alzeley ^,

Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 09 July 2026 Revised: 24 August 2026 Accepted: 04 September 2026 Published: 30 September 2026
MSC : 62M10, 62M05

This paper introduces a periodic logarithmic generalized autoregressive conditional heteroskedasticity (logGARCH) stochastic volatility model designed to capture volatility dynamics that evolve across different phases of a recurrent cycle. By allowing all model coefficients to vary periodically, the proposed framework extends the standard logGARCH stochastic volatility specification and provides a flexible mechanism for modeling seasonal variations and time-dependent persistence patterns in latent volatility processes. The probabilistic properties of the model are investigated through a periodic stochastic recurrence representation. Sufficient conditions are established for the existence of a unique strictly periodically stationary and causal solution. In addition, higher-order moment properties are derived, and explicit expressions for periodic moments are obtained under suitable integrability conditions. For statistical inference, the model is reformulated within a periodic state-space framework in which the latent log-volatility process is treated as an unobserved state variable. To estimate both the periodically varying parameters and the latent volatility states, a particle-assisted expectation-maximization algorithm combining sequential Monte Carlo filtering and smoothing techniques is developed. The finite-sample performance of the proposed methodology is evaluated through extensive Monte Carlo experiments under both Gaussian and Student-$ t_{15} $ innovations.
- stochastic volatility model,
- periodic $ \log $GARCH model,
- stationarity,
- EM algorithm,
- higher order moments,
- particle filtering
Citation: Omar Alzeley. Stationarity, moment properties, and particle-EM estimation for periodic logGARCH-SV models[J]. AIMS Mathematics, 2026, 11(9): 32199-32222. doi: 10.3934/math.20261266

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Abstract

This paper introduces a periodic logarithmic generalized autoregressive conditional heteroskedasticity (logGARCH) stochastic volatility model designed to capture volatility dynamics that evolve across different phases of a recurrent cycle. By allowing all model coefficients to vary periodically, the proposed framework extends the standard logGARCH stochastic volatility specification and provides a flexible mechanism for modeling seasonal variations and time-dependent persistence patterns in latent volatility processes. The probabilistic properties of the model are investigated through a periodic stochastic recurrence representation. Sufficient conditions are established for the existence of a unique strictly periodically stationary and causal solution. In addition, higher-order moment properties are derived, and explicit expressions for periodic moments are obtained under suitable integrability conditions. For statistical inference, the model is reformulated within a periodic state-space framework in which the latent log-volatility process is treated as an unobserved state variable. To estimate both the periodically varying parameters and the latent volatility states, a particle-assisted expectation-maximization algorithm combining sequential Monte Carlo filtering and smoothing techniques is developed. The finite-sample performance of the proposed methodology is evaluated through extensive Monte Carlo experiments under both Gaussian and Student-$ t_{15} $ innovations.

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