Estimating continuous-time bilinear process with moments method

Abstract

In the present paper, we study some probabilistic and statistical properties of a continuous-time version of the well-known bilinear processes driven by standard Brownian motion. This class of processes includes many popular processes defined as a nonlinear stochastic differential equation $\left( SDE\right) $ introduced \ recently in econometrics and statistics theory, because its capability to model many high-frequency data available in many fields particularly in mathematical finance. So, we examine firstly the probabilistic structure of the solution associated to continuous-time bilinear model $\left( COBL\right) $, its stationarity (resp. ergodicity) through the integrability on $\mathbb{R}_{+}$ of speed densities which we also briefly review. Secondly, the statistical properties of $COBL(1,1)$ are also discussed, and the explicit expressions of higher-order moments are derived. So, the first and second order moment we allow a direct method $\left( DM\right) $ to estimate the parameters. Moreover, based upon the discretization of the process over a finite time interval, we devote some
attention to an associated issue touching briefly the construction of moment method $\left( MM\right) $ estimation. However, we show that the parameter estimates obtained by $MM$ and $DM$ are strong consistency and asymptotically normal. A numerical \ simulation is presented confirming the performance of
$MM$ and $DM$ in the statistical inference of $SDE$.