Existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equations

The paper studies the existence of almost periodic solutions to some nonautonomous higher-order stochastic difference equation of the form:X\left( {t + n} \right) + \sum\limits_{r = 1}^{n - 1} {{A_r}\left( t \right)X\left( {t + r} \right) + {A_0}\left( t \right)X\left( t \right) = f\left( {t,X\left( t \right)} \right),}n ∈ 𝕑, by means of discrete dichotomy techniques.

Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations.In this work, we have studied the chaoticity properties of FIGARCH (p,d,q) processes by computing mutual information, correlation dimensions, FNNs (False Nearest Neighbour), the largest Lyapunov exponents (LLE) for both the stochastic difference equation and for the financial time series by applying Wolf’s algorithm, Kant’z algorithm and Jacobian algorithm. Although Wolf’s algorithm produced positive LLE’s, Kantz’s algorithm and Jacobian algorithm which are subsequently developed methods due to insufficiency of Wolf’s algorithm generated negative LLE’s constantly.So, as well as experimenting Wolf’s methods’ inefficiency formerly pointed out by Rosenstein (1993) and later Dechert and Gencay (2000), based on Kantz’s and Jacobian algorithm’s negative LLE outcomes, we concluded that it can be suggested that FIGARCH (p,d,q) is not deterministic chaotic process.

On studying relations between time series in climatology

In climatology, relationships between time series are often studied on the basis of crosscorrelation coefficients and regression equations. This approach is generally incorrect for time series irrespective of the crosscorrelation coefficient value because relations between time series are frequency-dependent. Multivariate time series should be analyzed in both time and frequency domains, including fitting a parametric (preferably, autoregressive) stochastic difference equation to the time series and then calculating functions of frequency such as spectra and coherent spectra, coherences, and frequency response functions. The example with a bivariate time series "Atlantic Multidecadal Oscillation (AMO) – sea surface temperature in Niño area 3.4 (SST3.4)" proves that even when the crosscorrelation is low, the time series' components can be closely related to each other. A full time and frequency domain description of this bivariate time series is given. The AMO − SST3.4 time series is shown to form a closed feedback loop system. The coherence between AMO and SST3.4 is statistically significant at intermediate frequencies where the coherent spectra amount up to 55% of the total spectral densities. The gain factors are also described. Some recommendations are offered regarding time series analysis in climatology.

Oscillation of Solutions to Second Order Stochastic Difference Equations

In this paper, we provide the criterion of existence of oscillation solutions to the following second order stochastic difference equation (please see in body part)

Quantifying and Reducing the Signal Noise Using Nonlinear Stochastic Difference Equations

This paper deals with quantification of the noise located in the digital signal by an innovative process. This process is associated with the filter that represents the novel information conveyed by the desired signal, residual interference and residual noise which are used to reduce the noise. A typical uniform quantization operation of a sampled signal is identified and interpreted with the framework of stochastic difference equation. A new theorem is proposed with all possible assumptions to support our result to signal noise ratio. AMS [200] Subject Classification: 39A10, 39A30, 39A60, 39B82, 39B99.