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

摘要

This paper is concerned with the investigation of $mathbb{L}_{2}$-structureissue of time-varying coefficients continuous-time bilinear processes ($COBL$)driven by a Brownian motion $left(BMight)$. Such processes are veryuseful for modeling irregular spacing non linear and non Gaussian datasets andmay be proposed to model for instance some financial returns representing highamplitude oscillations and thus make it a serious candidate for describeprocesses 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 theexistence of regular solutions in term of their transfer function. Expliciteformulas for the mean and covariance functions are given. As a consequence, weobserve that the second order structure is similar to a $CARMA$ processes withsome uncorrelated noise. Therefore, it is necessary to look intohigher-order cumulant in order to distinguish between $COBL$ and $CARMA$ processes.