A Note on $\mathbb{L}_{2}$-structure of Continuous-time Bilinear Processes with Time-varying Coefficients

Abstract

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.

The L2 – Structure of Subordinated Solution of Continuous-Time Bilinear Time Series

“…The existence of solution process of equation ( 1), was investigated by several authors, for instance, Iglói and Terdik [12] have studied the same model driven by fractional Brownian innovation. A class of COBL with time-varying coefficients was studied by Le Breton and Musiela [3], Bibi and Merahi [4] and Leon and Perez-Abreu [13]. Moreover, there are several monographic which discuss the theoretical probabilistic and statistical properties (interested readers are advised to see [14,15] and the references therein).…”

“…One of the classes of non-linear time-continuous models which has attracted considerable attention of the researchers is the classes of bilinear diffusion processes which have been widely studied and considered in time series analysis and in the theory of stochastic differential equations (SDE). For instance, among others, Le Breton and Musiela [3] and Bibi and Merahi [4] (1) in which μ x ðÞ¼αx þ μ and σ x ðÞ¼γx þ β are respectively the drift and diffusion functions representing respectively the conditional mean and variance of the infinitesimal change of X(t) at time t: wt ðÞ ðÞ t ≥ 0 is a real standard Brownian motion defined on some basic filtered space Ω, A, A t ðÞ t ≥ 0 , P ÀÁ and EXt ðÞ dw t ðÞ fg ¼ 0. The initial condition X(0) of X(t) can be either deterministic or random variable defined on Ω, A, P ðÞ independent of w such that EX0 ðÞ fg ¼ m 1 0 ðÞ and Var X 0 ðÞ fg ¼ K X 0 ðÞ .…”

GMM Estimation of Continuous-Time Bilinear Processes

“…The existence and uniqueness of the solution process of given by (see Le Breton and Musiela [16] and Bibi and Merahi [3])…”

“…The following theorem due to Bibi and Merahi [3], in which a recursive evolutionary transfer functions associated to the second-order regular solution of SDE (1) is given.…”