On the Goodness-of-Fit Tests for Some Continuous Time Processes

We present a review of several results concerning the construction of the Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size $\alpha$ and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.Comment: 22 pages, 2 figure

Estimation of cusp location of stochastic processes: a survey

© 2018, Springer Science+Business Media B.V., part of Springer Nature. We present a review of some recent results on estimation of location parameter for several models of observations with cusp-type singularity at the change point. We suppose that the cusp-type models fit better to the real phenomena described usually by change point models. The list of models includes Gaussian, inhomogeneous Poisson, ergodic diffusion processes, time series and the classical case of i.i.d. observations. We describe the properties of the maximum likelihood and Bayes estimators under some asymptotic assumptions. The asymptotic efficiency of estimators are discussed as well and the results of some numerical simulations are presented. We provide some heuristic arguments which demonstrate the convergence of log-likelihood ratios in the models under consideration to the fractional Brownian motion

On drift parameter estimation for mean-reversion type stochastic differential equations with discrete observations

We study the parameter estimation for mean-reversion type stochastic differential equations driven by Brownian motion. The equations, involving a small dispersion parameter, are observed at discrete (regularly spaced) time instants. The least square method is utilized to derive an asymptotically consistent estimator. Discussions on the rate of convergence of the least square estimator are presented. The new feature of this study is that, due to the mean-reversion type drift coefficient in the stochastic differential equations, we have to use the Girsanov transformation to simplify the equations, which then gives rise to the corresponding convergence of the least square estimator being with respect to a family of probability measures indexed by the dispersion parameter, while in the literature the existing results have dealt with convergence with respect to a given probability measure

Maximum Likelihood Estimator for Hidden Markov Models in continuous time

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.Comment: Warning: due to a flaw in the publishing process, some of the references in the published version of the article are confuse

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.Comment: 21 page

On limit distributions of estimators in irregular statistical models and a new representation of fractional Brownian motion

© 2018 Elsevier B.V. We provide new results concerning the limit distributions of Bayesian estimators (BE) and maximum likelihood estimators (MLE) of location parameters of cusp-type signals in “signal plus white noise” models. The limit distributions of BE and MLE are expressed in terms of fractional Brownian motion (fBm) with the Hurst parameter H, 0<H<1 as the noise intensity tends to zero. A new representation of fBm is given in terms of cusp functions. Simulation results for the densities and variances of the limit distributions of BE and MLE are also discussed

Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion

Stochastic differential equation, Trend, Nonparametric estimation, Kernel method, Small noise, Fractional Brownian motion, Primary 62M09, Secondary 60G15,