This paper is concerned with the investigation of L 2 -structure issue of time-varying coefficients continuous-time bilinear processes (COBL) driven by a Brownian motion (BM). Such processes are very useful for modeling irregular spacing non linear and non Gaussian datasets and may be proposed to model for instance some financial returns representing high amplitude oscillations and thus make it a serious candidate for describe processes with time-varying degree of persistence and other complex systems. Our attention is focused however on the probabilistic structure of COBL processes, so, we establish necessary and sufficient conditions for the existence of regular solutions in term of their transfer function. Explicite formulas for the mean and covariance functions are given. As a consequence, we observe that the second order structure is similar to a CARMA processes with some uncorrelated noise. Therefore, it is necessary to look into higher-order cumulant in order to distinguish between COBL and CARMA processes.
This paper examines the moments properties in frequency domain of the class of first order continuous-timebilinear processes (COBL(1,1) for short) with time-varying (resp. time-invariant) coefficients. So, we used theassociated evolutionary (or time-varying) transfer functions to study the structure of second-order of the process and its powers. In particular, for time-invariant case, an expression of the moments of any order are showed and the continuous-time AR (CAR) representation of COBL(1,1) is given as well as some moments properties of special cases. Based on these results we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM) illustrated by a Monte Carlo study and applied to modelling two foreign exchange rates of Algerian Dinar against U.S-Dollar (USD/DZD) and against the single European currency Euro (EUR/DZD).
<p style='text-indent:20px;'>Kernel functions play an important role in the complexity analysis of the interior point methods (IPMs) for linear optimization (LO). In this paper, an interior-point algorithm for LO based on a new parametric kernel function is proposed. By means of some simple analysis tools, we prove that the primal-dual interior-point algorithm for solving LO problems meets <inline-formula><tex-math id="M1">\begin{document}$ O\left(\sqrt{n} \log(n) \log(\frac{n}{\varepsilon}) \right) $\end{document}</tex-math></inline-formula>, iteration complexity bound for large-update methods with the special choice of its parameters.</p>
The continuous-time bilinear (COBL) process has been used to model non linear and/or non Gaussian datasets. In this paper, the first-order continuous-time bilinear COBL (1, 1) model driven by a fractional Brownian motion (fBm for short) process is presented. The use of fBm processes with certain Hurst parameter permits to obtain a much richer class of possibly long-range dependent property which are frequently observed in financial econometrics, and thus can be used as a power tool for modelling irregularly series having memory. So, the existence of Itô's solutions and there chaotic spectral representations for time-varying COBL (1, 1) processes driven by fBm are studied. The second-order properties of such solutions are analyzed and the long-range dependency property are studied.
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