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Note on local mixing techniques for stochastic differential equations

Alexander Veretennikov

https://doi.org/10.15559/21-VMSTA174

Pub. online: 16 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 1–15

Abstract

The paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). The coupling techniques traditionally consist of two components: one is local mixing, the other is recurrence. Often in the articles they do not split. Yet, they are quite different in their nature, and this paper separates them, concentrating only on the former.

Most of the techniques discussed here work in dimension $d\ge 1$, although, in $d=1$ there is one additional option to use intersections of trajectories, which requires nothing but the strong Markov property and nondegeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\dots $, or by arranging special stopping time sequences and to use the local Markov–Dobrushin (MD) condition, which is one of the most efficient versions of local mixing. Further applications may be based on one or another version of the MD condition; respectively, this paper is devoted to various methods of verifying one or another form of it.

Long-time behavior of a nonautonomous stochastic predator–prey model with jumps

Olga Borysenko Oleksandr Borysenko

https://doi.org/10.15559/21-VMSTA173

Pub. online: 8 Mar 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 17–39

Abstract

The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic predator–prey model with a modified version of the Leslie–Gower term and Holling-type II functional response disturbed by white noise, centered and noncentered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of a solution to the considered system.

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Olha Hopkalo Lyudmyla Sakhno

https://doi.org/10.15559/21-VMSTA171

Pub. online: 26 Jan 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 41–62

Abstract

This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/20-VMSTA74MI

Pub. online: 23 Dec 2020 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 473–476

Keywords index

https://doi.org/10.15559/20-VMSTA74KI

Pub. online: 23 Dec 2020 Type: Keywords Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 471–472

Author index

https://doi.org/10.15559/20-VMSTA74AI

Pub. online: 23 Dec 2020 Type: Author Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), p. 469

Averaging principle for a stochastic cable equation

Iryna Bodnarchuk

https://doi.org/10.15559/20-VMSTA168

Pub. online: 21 Dec 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 449–467

Abstract

We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.

Asymptotic normality of the residual correlogram in the continuous-time nonlinear regression model

Alexander Ivanov

Kateryna Moskvychova

https://doi.org/10.15559/20-VMSTA170

Pub. online: 21 Dec 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 93–113

Abstract

In a continuous time nonlinear regression model the residual correlogram is considered as an estimator of the stationary Gaussian random noise covariance function. For this estimator the functional central limit theorem is proved in the space of continuous functions. The result obtained shows that the limiting sample continuous Gaussian random process coincides with the limiting process in the central limit theorem for standard correlogram of the random noise in the specified regression model.

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

Luisa Beghin

Claudio Macci

Barbara Martinucci

https://doi.org/10.15559/20-VMSTA169

Pub. online: 21 Dec 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 1 (2021), pp. 63–91

Abstract

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. This kind of processes are useful in the study of chain molecular diffusions. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subordinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in [Journal of Statistical Physics 154 (2014), 1352–1364].

Asymptotic normality of modified LS estimator for mixture of nonlinear regressions

Vitalii Miroshnichenko Rostyslav Maiboroda

https://doi.org/10.15559/20-VMSTA167

Pub. online: 8 Dec 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 435–448

Abstract

We consider a mixture with varying concentrations in which each component is described by a nonlinear regression model. A modified least squares estimator is used to estimate the regressions parameters. Asymptotic normality of the derived estimators is demonstrated. This result is applied to confidence sets construction. Performance of the confidence sets is assessed by simulations.

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