We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.
We study the problem of parameter estimation for stochastic differential equations with small noise and fast oscillating parameters. Depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter, we consider three different regimes. For each regime, we construct the maximum likelihood estimator and we study its consistency and asymptotic normality properties. A simulation study for the first order Langevin equation with a two scale potential is also provided.
We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e. in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space-dependent and depend on multiple unknown parameters. It is known that classical estimators, such as Maximum Likelihood and Quadratic Variation of t…
This paper deals with the problems of consistence and strong consistence of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. A central limit theorem for these estimators is also obtained by using the Malliavin calculus.
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…
We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We sugge…
We construct the maximum likelihood estimator (MLE) of the unknown drift parameter $θ\in \mathbb{R}$ in the linear model $X_t=θt+σB^{H_1}(t)+B^{H_2}(t),\;t\in[0,T],$ where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $\frac12<H_1<H_2<1.$ The formula for MLE is based on the solution of the integral equation with weak polar kernel.
Consider an estimation of the Hurst parameter $H\in(0,1)$ and the volatility parameter $σ>0$ for a fractional Brownian motion with a drift term under high-frequency observations with a finite time interval. In the present paper, we propose a consistent estimator of the parameter $θ=(H,σ)$ combining the ideas of a quasi-likelihood function based on a local Gaussian approximation of a high-frequently observed time series and its frequency-domain approximation. Moreover, we prove an asymptotic normality property of the proposed estimator for all $H\in(0,1)$ when the drift process is constant.
Let $T>0,α>\frac12$. In the present paper we consider the $α$-Brownian bridge defined as $dX_t=-α\frac{X_t}{T-t}dt+dW_t,~ 0\leq t< T$, where $W$ is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $ α$ based on the continuous observation $\{X_s,0\leq s\leq t\}$ as $t\uparrow T$. We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $\frac{1}{\sqrt{|\log(T-t)|}}$, as $t\uparrow T$. First we compute an upper bound and then find a lower bound with the …
We study statistical inference for small-noise-perturbed multiscale dynamical systems. We prove consistency, asymptotic normality, and convergence of all scaled moments of an appropriately-constructed maximum likelihood estimator (MLE) for a parameter of interest, identifying precisely its limiting variance. We allow full dependence of coefficients on both slow and fast processes, which take values in the full Euclidean space; coefficients in the equation for the slow process need not be bounded and there is no assumption of periodic dependence. The results provide a theoretical basis for cali…
We consider a non-stationary Cox-Ingersoll-Ross process. We establish a sharp large deviation principle for the maximum likelihood estimator of its drift parameter.
We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We estimate simultaneously the drift and shift parameters. On the one hand, we establish a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the one previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood e…
We investigate the asymptotic properties of maximum likelihood estimators of the drift parameter for fractional vasicek model driven by a sub-fractional Brownian motion.
We consider the problem of finding anomalies in a $d$-dimensional field of independent random variables $\{Y_i\}_{i \in \left\{1,...,n\right\}^d}$, each distributed according to a one-dimensional natural exponential family $\mathcal F = \left\{F_θ\right\}_{θ\inΘ}$. Given some baseline parameter $θ_0 \inΘ$, the field is scanned using local likelihood ratio tests to detect from a (large) given system of regions $\mathcal{R}$ those regions $R \subset \left\{1,...,n\right\}^d$ with $θ_i \neq θ_0$ for some $i \in R$. We provide a unified methodology which controls the overall family wise error (FWE…
This paper is devoted to parameter estimation of the mixed fractional Ornstein-Uhlenbeck process with a drift. Large sample asymptotical properties of the Maximum Likelihood Estimator is deduced using the Laplace transform computations or the Cameron-Martin formula with extra part from \cite{CK19}
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…
We consider the problem of efficient estimation for the drift of fractional Brownian motion $B^H:=(B^H_t)_{t\in[0,T]}$ with hurst parameter $H$ less than 1/2. We also construct superefficient James-Stein type estimators which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.
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 consider a reflected Ornstein-Uhlenbeck process $X$ driven by a fractional Brownian motion with Hurst parameter $H\in (0, \frac12) \cup (\frac12, 1)$. Our goal is to estimate an unknown drift parameter $α\in (-\infty,\infty)$ on the basis of continuous observation of the state process. We establish Girsanov theorem for the process $X$, derive the standard maximum likelihood estimator of the drift parameter $α$, and prove its strong consistency and asymptotic normality. As an improved estimator, we obtain the explicit formulas for the sequential maximum likelihood estimator and its mean squa…
We consider statistical models driven by Gaussian and non-Gaussian self-similar processes with long memory and we construct maximum likelihood estimators (MLE) for the drift parameter. Our approach is based on the approximation by random walks of the driving noise. We study the asymptotic behavior of the estimators and we give some numerical simulations to illustrate our results.
Volume limitations and low yield thresholds of biological fluids have led to widespread use of passive microparticle rheology. The mean-squared-displacement (MSD) statistics of bead position time series (bead paths) are either applied directly to determine the creep compliance [Xu et al (1998)] or transformed to determine dynamic storage and loss moduli [Mason & Weitz (1995)]. A prevalent hurdle arises when there is a non-diffusive experimental drift in the data. Commensurate with the magnitude of drift relative to diffusive mobility, quantified by a Péclet number, the MSD statistics are disto…
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.
We consider the problem of efficient estimation of the drift parameter of an Ornstein-Uhlenbeck type process driven by a Lévy process when high-frequency observations are given. The estimator is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator and requires knowledge of the continuous martingale part. We use a thresholding technique to approximate the continuous part of the process. Under suitable conditions, we prove asymptotic normality and efficiency in the Hájek-Le Cam sense for the resulting drift estimator. Finally, we investi…
We refer by threshold Ornstein-Uhlenbeck to a continuous-time threshold autoregressive process. It follows the Ornstein-Uhlenbeck dynamics when above or below a fixed level, yet at this level (threshold) its coefficients can be discontinuous. We discuss (quasi)-maximum likelihood estimation of the drift parameters, both assuming continuous and discrete time observations. In the ergodic case, we derive consistency and speed of convergence of these estimators in long time and high frequency. Based on these results, we develop a test for the presence of a threshold in the dynamics. Finally, we ap…
The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consi…
We derive consistency and asymptotic normality results for quasi-maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous-time setting. The special feature of our analysis is that the stochastic integral part is unobserved and non-parametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi-parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results.
We establish large deviation principles for the couple of the maximum likelihood estimators of dimensional and drift coefficients in the generalised squared radial Ornstein-Uhlenbeck process. We focus our attention to the most tractable situation where the dimensional parameter $a>2$ and the drift parameter $b<0$. In contrast to the previous literature, we state large deviation principles when both dimensional and drift coefficient are estimated simultaneously.
We introduce a new method for high-dimensional, online changepoint detection in settings where a $p$-variate Gaussian data stream may undergo a change in mean. The procedure works by performing likelihood ratio tests against simple alternatives of different scales in each coordinate, and then aggregating test statistics across scales and coordinates. The algorithm is online in the sense that both its storage requirements and worst-case computational complexity per new observation are independent of the number of previous observations; in practice, it may even be significantly faster than this.…
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…
In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein-Uhlenbeck process which is defined as the solution of $dX_t=(L(t)-αX_t) dt + σdB_t$, and which is observed in continuous time. We derive an explicit representation of the generalized likelihood ratio test statistic assuming that the mean reversion function $L(t)$ is a finite linear combination of known basis functions. In the case of a periodic mean reversion function, we determine the asymptotic distribution of the test statistic under the null hypothesis.
We consider the problem of asymptotically efficient estimation of drift parameters of the ergodic fractional Ornstein-Uhlenbeck process under continuous observations when the Hurst parameter $H<1/2$ and the mean of its stationary distribution is not equal to zero. In this paper, we derive asymptotically efficient rates and variances of estimators of drift parameters and prove an asymptotic efficiency of a maximum likelihood estimator of drift parameters.
We propose HSMUCE (heterogeneous simultaneous multiscale change-point estimator) for the detection of multiple change-points of the signal in a heterogeneous gaussian regression model. A piecewise constant function is estimated by minimizing the number of change-points over the acceptance region of a multiscale test which locally adapts to changes in the variance. The multiscale test is a combination of local likelihood ratio tests which are properly calibrated by scale dependent critical values in order to keep a global nominal level alpha, even for finite samples. We show that HSMUCE control…
We study the problem of parametric estimation for continuously observed stochastic processes driven by additive small fractional Brownian motion with Hurst index 0<H<1/2 and 1/2<H<1. Under some assumptions on the drift coefficient, we obtain the asymptotic normality and moment convergence of maximum likelihood estimator of the drift parameter .
This paper deals with the rate of convergence for the central limit theorem of estimators of the drift coefficient, denoted $θ$, for a Ornstein-Uhlenbeck process $X \coloneqq \{X_t,t\geq0\}$ observed at high frequency. We provide an Approximate minimum contrast estimator and an approximate maximum likelihood estimator of $θ$, namely $\widetildeθ_{n}\coloneqq {1}/{\left(\frac{2}{n} \sum_{i=1}^{n}X_{t_{i}}^{2}\right)}$, and $\widehatθ_{n}\coloneqq -{\sum_{i=1}^{n} X_{t_{i-1}}\left(X_{t_{i}}-X_{t_{i-1}}\right)}/{\left(Δ_{n} \sum_{i=1}^{n} X_{t_{i-1}}^{2}\right)}$, respectively, where $ t_{i} = i …
We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian Lévy process of not necessarily bounded variation with a Lévy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. …
We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=αb(t)X_t dt + σ(t) dB_t$, $t\in[0,T)$, with a parameter $α\in R$, where $T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process. We formulate sufficient conditions under which the MLE of $α$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics. Next we study a SDE $dY_t=αb(t)a(Y_t) dt + σ(t) dB_t$, $t\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^γ)$ with some $γ\in[0,1)$. We give again sufficient co…
Stochastic differential equations and stochastic dynamics are good models to describe stochastic phenomena in real world. In this paper, we study N independent stochastic processes Xi(t) with real entries and the processes are determined by the stochastic differential equations with drift term relying on some random effects. We obtain the Girsanov-type formula of the stochastic differential equation driven by Fractional Brownian Motion through kernel transformation. Under some assumptions of the random effect, we estimate the parameter estimators by the maximum likelihood estimation and give s…
This paper deals with the estimation problem of misspecified ergodic Lévy driven stochastic differential equation models based on high-frequency samples. We utilize the widely applicable and tractable Gaussian quasi-likelihood approach which focuses on (conditional) mean and variance structure. It is shown that the corresponding Gaussian quasi-likelihood estimators of drift and scale parameters satisfy tail probability estimates and asymptotic normality at the same rate as correctly specified case. In this process, extended Poisson equation for time-homogeneous Feller Markov processes plays an…
We describe here a framework for a certain class of multiscale likelihood factorizations wherein, in analogy to a wavelet decomposition of an L^2 function, a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and the parameter vector localized in position and scale. The framework is developed as a set of sufficient conditions for the existence of such factorizations, formulated in analogy to those underlying a standard multiresolution analysis for wavelets, and hence can be viewed as a multiresolution analys…
We consider two transceivers, the first with perfect clock and the second with imperfect clock. We investigate the joint estimation of the delay between the transceivers and the offset and the drift of the imperfect clock. We propose a protocol for the synchronization of the clocks. We derive some empirical estimators for the delay, the offset and the drift, and compute the Cramer-Rao lower bounds and the joint maximum likelihood estimator of the delay and the drift. We study the impact of the protocol parameters and the time-of-arrival estimation variance on the achieved performances. We vali…
We propose a method for obtaining maximum likelihood estimates (MLEs) of a Markov-Modulated Jump-Diffusion Model (MMJDM) when the data is a discrete time sample of the diffusion process, the jumps follow a Laplace distribution, and the parameters of the diffusion are controlled by a Markov Jump Process (MJP). The data can be viewed as incomplete observation of a model with a tractable likelihood function. Therefore we use the EM-algorithm to obtain MLEs of the parameters. We validate our method with simulated data. The motivation for obtaining estimates of this model is that stock prices have …
We consider adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones for discretely observed ergodic diffusion processes with observation noise whose variance is constant. The quasi-likelihood functions for the diffusion and drift parameters are introduced and the polynomial-type large deviation inequalities for those quasi-likelihoods are shown to see the convergence of moments for those estimators.
In many applications it is desirable to infer coarse-grained models from observational data. The observed process often corresponds only to a few selected degrees of freedom of a high-dimensional dynamical system with multiple time scales. In this work we consider the inference problem of identifying an appropriate coarse-grained model from a single time series of a multiscale system. It is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting. Here we present a novel parametric inference methodolo…
Due to their conjugate posteriors, Gaussian process priors are attractive for estimating the drift of stochastic differential equations with continuous time observations. However, their performance strongly depends on the choice of the hyper-parameters. We employ the marginal maximum likelihood estimator to estimate the scaling and/or smoothness parameter(s) of the prior and show that the corresponding posterior has optimal rates of convergence. General theorems do not apply directly to this model as the usual test functions are with respect to a random Hellinger-type metric. We allow for cont…
