Test for parameter change in discretely observed diffusion processes

In this paper, we consider the problem of testing for the parameter change in discretely observed diffusion processes. For a test, we perform the cusum test based on the estimator proposed by Kessler (J Stat 24:211–229, 1997). It is shown that the test statistic weakly converges to the sup of the square of a Browian bridge. Simulation results are illustrated for the Ornstein-Uhlenbeck and Cox-Ingersoll-Ross processes.     

Uniform concentration inequality for ergodic diffusion processes observed at discrete times

In this paper a concentration inequality is proved for the deviation in the ergodic theorem for diffusion processes in the case of discrete time observations. The proof is based on geometric ergodicity of diffusion processes. We consider as an application the nonparametric pointwise estimation problem of the drift coefficient when the process is observed at discrete times.     

Estimation of parameters for discretely observed diffusion processes with a variety of rates for information

A specific form of stochastic differential equation with unknown parameters are considered. We do not necessarily assume ergodicity or recurrency, and any moment conditions for the true process, but some tightness conditions for an information-like quantity. The interest is to estimate the parameters from discrete observations the step size of which tends to zero. Consistency and the rate of convergence of proposed estimators are presented. The rate is deduced naturally from the rate for the information-like quantities.     

Estimation for discretely observed continuous state branching processes with immigration

We study the problem of parameter estimation for the continuous state branching processes with immigration, observed at discrete time points. The weighted conditional least square estimators (WCLSEs) are used for the drift parameters. Under the proper moment conditions, asymptotic distributions of the WCLSEs are obtained in the supercritical, sub- or critical cases.     

Asymptotic equivalence of discretely observed diffusion processes and their Euler scheme: small variance case

This paper establishes the global asymptotic equivalence, in the sense of the Le Cam \(\Delta \)-distance, between scalar diffusion models with unknown drift function and small variance on the one side, and nonparametric autoregressive models on the other side. The time horizon \(T\) is kept fixed and both the cases of discrete and continuous observation of the path are treated. We allow non constant diffusion coefficient, bounded but possibly tending to zero. The asymptotic equivalences are established by constructing explicit equivalence mappings.     

Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations

We consider adaptive Bayesian estimation of both drift and diffusion coefficient parameters for ergodic multidimensional diffusion processes based on sampled data. Under a general condition on the discretization step of the sampled data, three kinds of adaptive Bayes type estimators are proposed by applying adaptive maximum likelihood type methods of Uchida and Yoshida (Stoch Process Appl 122:2885–2924, 2012) to Bayesian procedures. We show asymptotic normality and convergence of moments for the adaptive Bayes type estimators by means of the Ibragimov–Has’minskii–Kutoyants program together with the polynomial type large deviation inequality for the statistical random field.     

Goodness of fit test for ergodic diffusion processes

A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypothesis and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact rejection regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.     

Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(α,_Xt)$b(\alpha ,X_t)$ and diffusion coefficient ?σ(β,_Xt)$?\sigma (\beta ,X_t)$. The diffusion sample path is discretely observed at times _tk=kΔ$t_k=k\Delta$ for k=1…n$k=1\dots n$ on a fixed interval [0,T]$[0,T]$. We study minimum contrast estimators derived from the Gaussian process approximating X$X$ for small ?$?$. We obtain consistent and asymptotically normal estimators of α$\alpha$ for fixed Δ$\Delta$ and ?→0$?\to 0$ and of (α,β)$(\alpha ,\beta )$ for Δ→0$\Delta \to 0$ and ?→0$?\to 0$ without any condition linking ?$?$ and Δ$\Delta$. We compare the estimators obtained with various methods and for various magnitudes of Δ$\Delta$ and ?$?$ based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.     

Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes

The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against “presence of a jump component” and “diffusion-coefficient misspecification”; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the correctly specified diffusion type model. Especially, our asymptotic result clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator.     

Bayesian ergodic adaptive control of diffusion processes

A Bayesian adaptive control problem for ergodic diffusions is considered. A nearly self-optimising strategy is constructed on the basis of discrete-time observations of the state process. In this derivation a key role is played by an embedded Markov chain with an associated Bellman-type equation which appears to be of independent interest     

Asymptotic normality for discretely observed Markov jump processes with an absorbing state

For a continuous-time Markov process, occasionally, only discrete-time observations are available. For a simple sample of homogeneous Markov jump processes with an absorbing state, observed each on a stochastic grid of time points, we establish asymptotic normality of the maximum likelihood estimator and close the gap in Kremer and Weißbach (2013). By showing that the solution of the Kolmogorov backward equation system is continuous differentiable, we can apply results for M-estimators.     

Quasi-likelihood analysis for nonsynchronously observed diffusion processes

We consider nonsynchronous sampling of parameterized stochastic regression models, which contain stochastic differential equations. Constructing a quasi-likelihood function, we prove that the quasi-maximum likelihood estimator and the Bayes type estimator are consistent and asymptotically mixed normal when the sampling frequency of the nonsynchronous data becomes large.     

Contrast-based information criterion for ergodic diffusion processes from discrete observations

In this paper, we consider the model selection problem for discretely observed ergodic multi-dimensional diffusion processes. In order to evaluate the statistical models, Akaike’s information criterion (AIC) is a useful tool. Since AIC is constructed by the maximum log likelihood and the dimension of the parameter space, it may look easy to get AIC even for discretely observed diffusion processes. However, there is a serious problem that a transition density of a diffusion process does not generally have an explicit form. Instead of the exact log-likelihood, we use a contrast function based on a locally Gaussian approximation of the transition density and we propose the contrast-based information criterion.     

On asymptotic distribution of parameter free tests for ergodic diffusion processes

We consider two problems of constructing of goodness of fit tests for ergodic diffusion processes. The first one is concerned with a composite basic hypothesis for a parametric class of diffusion processes, which includes the Ornstein–Uhlenbeck and simple switching processes. In this case we propose asymptotically parameter free tests of Cramér-von Mises type. The basic hypothesis in the second problem is simple and we propose asymptotically distribution free tests for a wider class of trend coefficients.     

Parametric inference for discretely observed subordinate diffusions

Statistical Inference for Stochastic Processes - Subordinate diffusions are constructed by time changing diffusion processes with an independent Lévy subordinator. This is a rich family of...     

On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes

We consider the problem of the construction of the asymptotically distribution free test by the observations of ergodic diffusion process. It is supposed that under the basic hypothesis the trend coefficient depends on a finite-dimensional parameter and we study the Cramér-von Mises type statistics. The underlying statistics depends on the deviation of the local time estimator from the invariant density with parameter replaced by the maximum likelihood estimator. We propose a linear transformation which yields the convergence of the test statistics to an integral of the Wiener process. Therefore the test based on this statistics is asymptotically distribution free.     

Optimal ergodic control of Markov diffusion processes with minimum variance

In this paper, we study the optimal ergodic control problem with minimum variance for a general class of controlled Markov diffusion processes. To this end, we follow a lexicographical approach. Namely, we first identify the class of average optimal control policies, and then within this class, we search policies that minimize the limiting average variance. To do this, a key intermediate step is to show that the limiting average variance is a constant independent of the initial state. Our proof of this latter fact gives a result stronger than the central limit theorem for diffusions. An application to manufacturing systems illustrates our results.