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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.

Linear backward stochastic differential equations with Gaussian Volterra processes

Habiba Knani Marco Dozzi

https://doi.org/10.15559/20-VMSTA166

Pub. online: 3 Dec 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 415–433

Abstract

Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional Brownian motion and the multifractional Ornstein-Uhlenbeck process. By an Itô formula, proven in the context of Malliavin calculus, the BSDE is associated to a linear second order partial differential equation with terminal condition whose solution is given by a Feynman-Kac type formula.

Subordinated compound Poisson processes of order k

Ayushi Singh Sengar Neelesh S. Upadhye

https://doi.org/10.15559/20-VMSTA165

Pub. online: 5 Nov 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 395–413

Abstract

In this article, the compound Poisson process of order k (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinators (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that the space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the process defined in [Statist. Probab. Lett. 82 (2012), 852–858].

On shortfall risk minimization for game options

Yan Dolinsky

https://doi.org/10.15559/20-VMSTA164

Pub. online: 29 Oct 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 379–394

Abstract

In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black–Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist.

Geometric branching reproduction Markov processes

Assen Tchorbadjieff Penka Mayster

https://doi.org/10.15559/20-VMSTA163

Pub. online: 30 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 357–378

Abstract

We present a model of a continuous-time Markov branching process with the infinitesimal generating function defined by the geometric probability distribution. It is proved that the solution of the backward Kolmogorov equation is expressed by the composition of special functions – Wright function in the subcritical case and Lambert-W function in the critical case. We found the explicit form of conditional limit distribution in the subcritical branching reproduction. In the critical case, the extinction probability and probability mass function are expressed as a series containing Bell polynomial, Stirling numbers, and Lah numbers.

Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation

Diana Avetisian Kostiantyn Ralchenko

https://doi.org/10.15559/20-VMSTA162

Pub. online: 18 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 339–356

Abstract

The paper deals with a stochastic heat equation driven by an additive fractional Brownian space-only noise. We prove that a solution to this equation is a stationary and ergodic Gaussian process. These results enable us to construct a strongly consistent estimator of the diffusion parameter.

On tail behaviour of stationary second-order Galton–Watson processes with immigration

Mátyás Barczy

Zsuzsanna Bősze Gyula Pap

https://doi.org/10.15559/20-VMSTA161

Pub. online: 10 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 315–338

Abstract

Sufficient conditions are presented on the offspring and immigration distributions of a second-order Galton–Watson process ${({X_{n}})_{n\geqslant -1}}$ with immigration, under which the distribution of the initial values $({X_{0}},{X_{-1}})$ can be uniquely chosen such that the process becomes strongly stationary and the common distribution of ${X_{n}}$, $n\geqslant -1$, is regularly varying.

On distributions of exponential functionals of the processes with independent increments

Lioudmila Vostrikova

https://doi.org/10.15559/20-VMSTA159

Pub. online: 8 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 291–313

Abstract

The aim of this paper is to study the laws of exponential functionals of the processes $X={({X_{s}})_{s\ge 0}}$ with independent increments, namely

\[ {I_{t}}={\int _{0}^{t}}\exp (-{X_{s}})ds,\hspace{0.1667em}\hspace{0.1667em}t\ge 0,\]

and also

\[ {I_{\infty }}={\int _{0}^{\infty }}\exp (-{X_{s}})ds.\]

Under suitable conditions, the integro-differential equations for the density of ${I_{t}}$ and ${I_{\infty }}$ are derived. Sufficient conditions are derived for the existence of a smooth density of the laws of these functionals with respect to the Lebesgue measure. In the particular case of Lévy processes these equations can be simplified and, in a number of cases, solved explicitly.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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