Latest articles of Modern Stochastics: Theory and Applications
http://vmsta.org/journal/VMSTA/feeds/latest
https://vmsta.org/https://vmsta.org/Latest articles of Modern Stochastics: Theory and Applications
http://vmsta.org/journal/VMSTA/feeds/latest
enThu, 02 Feb 2023 17:49:04 +0200<![CDATA[Parameter estimation in mixed fractional stochastic heat equation]]>
https://vmsta.org/journal/VMSTA/article/263
https://vmsta.org/journal/VMSTA/article/263The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3/4$. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.
PDFXML]]>The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3/4$. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.
PDFXML]]>Diana Avetisian,Kostiantyn RalchenkoTue, 24 Jan 2023 00:00:00 +0200<![CDATA[A modified Φ-Sobolev inequality for canonical Lévy processes and its applications]]>
https://vmsta.org/journal/VMSTA/article/262
https://vmsta.org/journal/VMSTA/article/262A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.
PDFXML]]>A new modified Φ-Sobolev inequality for canonical ${L^{2}}$-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert’s W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes.
PDFXML]]>Noriyoshi Sakuma,Ryoichi SuzukiMon, 23 Jan 2023 00:00:00 +0200<![CDATA[Some examples of noncentral moderate deviations for sequences of real random variables]]>
https://vmsta.org/journal/VMSTA/article/261
https://vmsta.org/journal/VMSTA/article/261The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.
PDFXML]]>The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered normal distribution. In this paper, some examples of classes of large deviation principles of this kind are presented, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.
PDFXML]]>Rita Giuliano,Claudio MacciThu, 19 Jan 2023 00:00:00 +0200<![CDATA[Reflected generalized discontinuous BSDEs with rcll barrier and an obstacle problem of IPDE with nonlinear Neumann boundary conditions]]>
https://vmsta.org/journal/VMSTA/article/260
https://vmsta.org/journal/VMSTA/article/260Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.
PDFXML]]>Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.
PDFXML]]>Mohammed Elhachemy,Mohamed El OtmaniFri, 30 Dec 2022 00:00:00 +0200<![CDATA[Lévy processes conditioned to stay in a half-space with applications to directional extremes]]>
https://vmsta.org/journal/VMSTA/article/259
https://vmsta.org/journal/VMSTA/article/259This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.
PDFXML]]>This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.
PDFXML]]>Jevgenijs Ivanovs,Jakob D. ThøstesenFri, 25 Nov 2022 00:00:00 +0200<![CDATA[A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution]]>
https://vmsta.org/journal/VMSTA/article/258
https://vmsta.org/journal/VMSTA/article/258We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.
PDFXML]]>We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.
PDFXML]]>Luigi Amedeo Bianchi,Stefano Bonaccorsi,Luciano TubaroMon, 21 Nov 2022 00:00:00 +0200<![CDATA[Keywords index]]>
https://vmsta.org/journal/VMSTA/article/256
https://vmsta.org/journal/VMSTA/article/256PDF XML]]>PDF XML]]>Wed, 09 Nov 2022 00:00:00 +0200<![CDATA[2010 Mathematics Subject Classification index]]>
https://vmsta.org/journal/VMSTA/article/257
https://vmsta.org/journal/VMSTA/article/257PDF XML]]>PDF XML]]>Wed, 09 Nov 2022 00:00:00 +0200<![CDATA[Author index]]>
https://vmsta.org/journal/VMSTA/article/255
https://vmsta.org/journal/VMSTA/article/255PDF XML]]>PDF XML]]>Wed, 09 Nov 2022 00:00:00 +0200<![CDATA[Minimax identity with robust utility functional for a nonconcave utility]]>
https://vmsta.org/journal/VMSTA/article/254
https://vmsta.org/journal/VMSTA/article/254The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.
PDFXML]]>The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.
PDFXML]]>Olena Bahchedjioglou,Georgiy ShevchenkoFri, 28 Oct 2022 00:00:00 +0300