We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(α-βX_t)\,dt+γ\,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $α$ and $β$ in the non-ergodic case (when $β<0$) for arbitrary $x_0\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=α/β$, derive their asymptotic distributions and prove their asymptotic independence.
We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $dX_t=(μ+θX_t)dt+dG_t,\ t\geq0$ with unknown parameters $θ>0$ and $μ\in\mathbb{R}$, where $G$ is a Gaussian process. We provide least square-type estimators $\widetildeθ_T$ and $\widetildeμ_T$ respectively for the drift parameters $θ$ and $μ$ based on continuous-time observations $\{X_t,\ t\in[0,T]\}$ as $T\rightarrow\infty$. Our aim is to derive some sufficient conditions on the driving Gaussian process $G$ in order to ensure that $\widetildeθ_T$ and $\widetildeμ_T$ are strongly consistent, t…
We investigate the asymptotic properties of maximum likelihood estimators of the drift parameter for fractional vasicek model driven by a sub-fractional Brownian motion.
The mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model, has been widely used in modelling volatility, interest rate and exchange rate. Obviously, if some phenomenon are modeled by the mixed fractional Vasicek model, statistical inference for this process is of great interest. Based on continuous time observations, this paper considers the problem of estimating the drift parameters in the mixed fractional Vasicek model. We will propose the maximum likelihood estimators of the drift parameters in the mixed fractional Vasicek model with the Radon-Nikodym d…
The class of observation-driven models (ODMs) includes many models of non-linear time series which, in a fashion similar to, yet different from, hidden Markov models (HMMs), involve hidden variables. Interestingly, in contrast to most HMMs, ODMs enjoy likelihoods that can be computed exactly with computational complexity of the same order as the number of observations, making maximum likelihood estimation the privileged approach for statistical inference for these models. A celebrated example of general order ODMs is the GARCH$(p,q)$ model, for which ergodicity and inference has been studied e…
We consider a discrete-time Markov chain $\boldsymbolΦ$ on a general state-space ${\sf X}$, whose transition probabilities are parameterized by a real-valued vector $\boldsymbolθ$. Under the assumption that $\boldsymbolΦ$ is geometrically ergodic with corresponding stationary distribution $π(\boldsymbolθ)$, we are interested in estimating the gradient $\nabla α(\boldsymbolθ)$ of the steady-state expectation $$α(\boldsymbolθ) = π( \boldsymbolθ) f.$$ To this end, we first give sufficient conditions for the differentiability of $α(\boldsymbolθ)$ and for the calculation of its gradient via a seque…
We establish a moderate deviation principle for the maximum likelihood estimator of the four parameters of a geometrically ergodic Heston process. We also obtain moderate deviations for the maximum likelihood estimator of the couple of dimensional and drift parameters of a generalized squared radial Ornstein-Uhlenbeck process. We restrict ourselves to the most tractable case where the dimensional parameter satisfies $a>2$ and the drift coefficient is such that $b<0$. In contrast to the previous literature, parameters are estimated simultaneously.
The aim of this paper is to introduce a new type of test statistic for simple null hypothesis on one-dimensional ergodic diffusion processes sampled at discrete times. We deal with a quasi-likelihood approach for stochastic differential equations (i.e. local gaussian approximation of the transition functions) and define a test statistic by means of the empirical $L^2$-distance between quasi-likelihoods. We prove that the introduced test statistic is asymptotically distribution free; namely it weakly converges to a $χ^2$ random variable. Furthermore, we study the power under local alternatives …
We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In lon…
The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to I.Ibragimov and R.Khasminskii, consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity conditions of the chain.
Gaussian quasi-likelihood estimation of the parameter $θ$ in the square-root diffusion process is studied under high frequency sampling. Different from the previous study of Overbeck and Rydén(1998) under low-frequency sampling, high-frequency of data provides very simple form of the asymptotic covariance matrix. Through easy-to-compute preliminary contrast functions, a practical two-stage manner without numerical optimization is formulated in order to conduct not only an asymptotically efficient estimation of the drift parameters, but also high-precision estimator of the diffusion parameter. …
In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that ex…
We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1/2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate $\sqrtτ$ as $τ\rightarrow \infty$, when the solution is observed continuously on the time interval $[0,τ]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyse the particular case of the fractional Ornstein-Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in th…
Multidimensional hypoelliptic diffusions arise naturally in different fields, for example to model neuronal activity. Estimation in those models is complex because of the degenerate structure of the diffusion coefficient. In this paper we consider hypoelliptic diffusions, given as a solution of two-dimensional stochastic differential equations (SDEs), with the discrete time observations of both coordinates being available on an interval $T = nΔ_n$, with $Δ_n$ the time step between the observations. The estimation is studied in the asymptotic setting, with $T\to\infty$ as $Δ_n\to 0$. We build a…
A number of discrete time, finite population size models in genetics describing the dynamics of allele frequencies are known to converge (subject to suitable scaling) to a diffusion process in the infinite population limit, termed the Wright-Fisher diffusion. In this article we show that the diffusion is ergodic uniformly in the selection and mutation parameters, and that the measures induced by the solution to the stochastic differential equation are uniformly locally asymptotically normal. Subsequently these two results are used to analyse the statistical properties of the Maximum Likelihood…
Likelihood-based inference for birth-death processes with composite birth mechanisms
Network Science Institute, Northeastern University London（伦敦东北大学网络科学研究所）；Department of Mathematics, University of Sussex（萨塞克斯大学数学系）；Department of Mathematics, Northeastern University（东北大学数学系）
The objective of the paper is to identify and investigate all possible types of asymptotic behavior for the maximum likelihood estimators of the unknown parameters in the second-order linear stochastic ordinary differential equation driven by Gaussian white noise. The emphasis is on the non-ergodic case, when the roots of the corresponding characteristic equation are not both in the left half-plane.
We revisit the problem of estimating the parameters of a partially observed diffusion process, consisting of a hidden state process and an observed process, with a continuous time parameter. The estimation is to be done online, i.e. the parameter estimate should be updated recursively based on the observation filtration. We provide a theoretical analysis of the stochastic gradient ascent algorithm on the incomplete-data log-likelihood. The convergence of the algorithm is proved under suitable conditions regarding the ergodicity of the process consisting of state, filter, and tangent filter. Ad…
In the presence of multiscale dynamics in a reaction network, direct simulation methods become inefficient as they can only advance the system on the smallest scale. This work presents stochastic averaging techniques to accelerate computations for obtaining estimates of expected values and sensitivities with respect to the steady state distribution. A two-time-scale formulation is used to establish bounds on the bias induced by the averaging method. Further, this formulation provides a framework to create an accelerated `averaged' version of most single-scale sensitivity estimation method. In …
We consider the problem of estimation of the drift parameter of an ergodic Ornstein--Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed continuously on a long time interval $[0,T]$, $T\to\infty$. We prove that the statistical model is locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically efficient.
We seek to infer the parameters of an ergodic Markov process from samples taken independently from the steady state. Our focus is on non-equilibrium processes, where the steady state is not described by the Boltzmann measure, but is generally unknown and hard to compute, which prevents the application of established equilibrium inference methods. We propose a quantity we call propagator likelihood, which takes on the role of the likelihood in equilibrium processes. This propagator likelihood is based on fictitious transitions between those configurations of the system which occur in the sample…
We consider the problem of parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We study two cases: one in which we observe multiple independent trajectories of the McKean-Vlasov SDE, and another in which we observe multiple particles from the interacting particle system. In each case, we begin by establishing consistency and asymptotic normality of the (approximate) offline maximum likelihood estimator, in the limit as the number of observations $N\rightarrow\infty$. We then propose an online maximum likelihood estimator, whi…
