In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $m+1$, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.

This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method.

This paper explored a new three-dimensional nonlinear system of difference equations, capturing intricate dynamic interactions through advanced analytical and computational methods. By employing strategic transformations and analyzing the system's characteristic polynomial roots, we derived exact closed-form solutions for both distinct and repeated root scenarios. Numerical examples were used to validate the theoretical results and demonstrate that even small variations in initial conditions can induce markedly different dynamical behaviors, ranging from stable oscillations to divergent trajectories.