We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter $theta_1$ in a non-degenerate diffusion coefficient and a parameter $theta_2$ in
the drift term. The second component has a drift term parameterized by $theta_3$ and no diffusion term. Asymptotic normality is proved in three different situations for an adaptive estimator for $theta_3$ with some initial estimators for ($theta_1$ , $theta_2$), an adaptive one-step estimator for ($theta_1$ , $theta_2$ , $theta_3$) with some initial estimators for them, and a joint quasi-maximum likelihood estimator for ($theta_1$ , $theta_2$ , $theta_3$) without any initial estimator. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for $theta_1$ is smaller than the standard one based only on the first component. The convergence of the estimators for $theta_3$ is much faster than the other parameters. The resulting asymptotic variance is smaller than that of an estimator only using the increments of the second component.
We study the problem of the non-parametric estimation for the density $pi$ of the stationary distribution of a stochastic two-dimensional damping Hamiltonian system $(Z_t)_{tin[0,T]}=(X_t,Y_t)_{t in [0,T]}$. From the continuous observation of the sam
pling path on $[0,T]$, we study the rate of estimation for $pi(x_0,y_0)$ as $T to infty$. We show that kernel based estimators can achieve the rate $T^{-v}$ for some explicit exponent $v in (0,1/2)$. One finding is that the rate of estimation depends on the smoothness of $pi$ and is completely different with the rate appearing in the standard i.i.d. setting or in the case of two-dimensional non degenerate diffusion processes. Especially, this rate depends also on $y_0$. Moreover, we obtain a minimax lower bound on the $L^2$-risk for pointwise estimation, with the same rate $T^{-v}$, up to $log(T)$ terms.
We introduce a Cox-type model for relative intensities of orders flows in a limit order book. The model assumes that all intensities share a common baseline intensity, which may for example represent the global market activity. Parameters can be esti
mated by quasi likelihood maximization, without any interference from the baseline intensity. Consistency and asymptotic behavior of the estimators are given in several frameworks, and model selection is discussed with information criteria and penalization. The model is well-suited for high-frequency financial data: fitted models using easily interpretable covariates show an excellent agreement with empirical data. Extensive investigation on tick data consequently helps identifying trading signals and important factors determining the limit order book dynamics. We also illustrate the potential use of the framework for out-of-sample predictions.
We propose a parametric model for the simulation of limit order books. We assume that limit orders, market orders and cancellations are submitted according to point processes with state-dependent intensities. We propose new functional forms for these
intensities, as well as new models for the placement of limit orders and cancellations. For cancellations, we introduce the concept of priority index to describe the selection of orders to be cancelled in the order book. Parameters of the model are estimated using likelihood maximization. We illustrate the performance of the model by providing extensive simulation results, with a comparison to empirical data and a standard Poisson reference.
