Latest articles of Modern Stochastics: Theory and Applications
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https://www.vmsta.org/https://www.vmsta.org/Latest articles of Modern Stochastics: Theory and Applications
http://www.vmsta.org/journal/VMSTA/feeds/latest
enFri, 20 May 2022 03:28:14 +0300<![CDATA[Gaussian Volterra processes with power-type kernels. Part I]]>
https://www.vmsta.org/journal/VMSTA/article/244
https://www.vmsta.org/journal/VMSTA/article/244The stochastic process of the form

is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$ on the entire interval $[0,T]$.
PDFXML]]>The stochastic process of the form

is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$ on the entire interval $[0,T]$.
PDFXML]]>Yuliya Mishura,Sergiy ShklyarWed, 27 Apr 2022 00:00:00 +0300<![CDATA[Spatial birth-and-death processes with a finite number of particles]]>
https://www.vmsta.org/journal/VMSTA/article/243
https://www.vmsta.org/journal/VMSTA/article/243The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over ${\mathbb{R}^{\mathrm{d}}}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator.

The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

PDFXML]]>The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over ${\mathbb{R}^{\mathrm{d}}}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator.

The pathwise behavior of an aggregation model is also studied. The probability of extinction and the growth rate of the number of particles under condition of nonextinction are estimated.

PDFXML]]>Viktor Bezborodov,Luca Di PersioTue, 19 Apr 2022 00:00:00 +0300<![CDATA[On the denseness of the subset of discrete distributions in a certain set of two-dimensional distributions]]>
https://www.vmsta.org/journal/VMSTA/article/242
https://www.vmsta.org/journal/VMSTA/article/242In the article [Theory of Probability & Its Applications 62(2) (2018), 216–235], a class $\mathbb{W}$ of terminal joint distributions of integrable increasing processes and their compensators was introduced. In this paper, it is shown that the discrete distributions lying in $\mathbb{W}$ form a dense subset in the set $\mathbb{W}$ for ψ-weak topology with a gauge function ψ of linear growth.
PDFXML]]>In the article [Theory of Probability & Its Applications 62(2) (2018), 216–235], a class $\mathbb{W}$ of terminal joint distributions of integrable increasing processes and their compensators was introduced. In this paper, it is shown that the discrete distributions lying in $\mathbb{W}$ form a dense subset in the set $\mathbb{W}$ for ψ-weak topology with a gauge function ψ of linear growth.
PDFXML]]>Dmitriy Borzykh,Alexander GushchinFri, 25 Mar 2022 00:00:00 +0200<![CDATA[Random walks with sticky barriers]]>
https://www.vmsta.org/journal/VMSTA/article/239
https://www.vmsta.org/journal/VMSTA/article/239A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.
PDFXML]]>A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.
PDFXML]]>Vladyslav Bohun,Alexander MarynychWed, 16 Mar 2022 00:00:00 +0200<![CDATA[Factorial moments of the critical Markov branching process with geometric reproduction of particles]]>
https://www.vmsta.org/journal/VMSTA/article/238
https://www.vmsta.org/journal/VMSTA/article/238The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.
PDFXML]]>The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.
PDFXML]]>Assen Tchorbadjieff,Penka MaysterMon, 07 Feb 2022 00:00:00 +0200<![CDATA[Asymptotic results for families of random variables having power series distributions]]>
https://www.vmsta.org/journal/VMSTA/article/235
https://www.vmsta.org/journal/VMSTA/article/235Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].
PDFXML]]>Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].
PDFXML]]>Claudio Macci,Barbara Pacchiarotti,Elena VillaThu, 03 Feb 2022 00:00:00 +0200<![CDATA[Conditional LQ time-inconsistent Markov-switching stochastic optimal control problem for diffusion with jumps]]>
https://www.vmsta.org/journal/VMSTA/article/236
https://www.vmsta.org/journal/VMSTA/article/236The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.
PDFXML]]>The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.
PDFXML]]>Nour El Houda Bouaicha,Farid Chighoub,Ishak Alia,Ayesha SohailThu, 03 Feb 2022 00:00:00 +0200<![CDATA[Models of space-time random fields on the sphere]]>
https://www.vmsta.org/journal/VMSTA/article/237
https://www.vmsta.org/journal/VMSTA/article/237General models of random fields on the sphere associated with nonlocal equations in time and space are studied. The properties of the corresponding angular power spectrum are discussed and asymptotic results in terms of random time changes are found.
PDFXML]]>General models of random fields on the sphere associated with nonlocal equations in time and space are studied. The properties of the corresponding angular power spectrum are discussed and asymptotic results in terms of random time changes are found.
PDFXML]]>Mirko D’Ovidio,Enzo Orsingher,Lyudmyla SakhnoThu, 03 Feb 2022 00:00:00 +0200<![CDATA[Averaging principle for the one-dimensional parabolic equation driven by stochastic measure]]>
https://www.vmsta.org/journal/VMSTA/article/233
https://www.vmsta.org/journal/VMSTA/article/233A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure is considered. The averaging principle for the equation is established. The convergence rate is compared with other results on related topics.
PDFXML]]>A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure is considered. The averaging principle for the equation is established. The convergence rate is compared with other results on related topics.
PDFXML]]>Boris ManikinMon, 10 Jan 2022 00:00:00 +0200<![CDATA[Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices]]>
https://www.vmsta.org/journal/VMSTA/article/234
https://www.vmsta.org/journal/VMSTA/article/234The generalised sine random point field arises from the scaling limit at the origin of the eigenvalues of the generalised Gaussian ensembles. We solve an infinite-dimensional stochastic differential equation (ISDE) describing an infinite number of interacting Brownian particles which is reversible with respect to the generalised sine random point field. Moreover, finite particle approximation of the ISDE is shown, that is, a solution to the ISDE is approximated by solutions to finite-dimensional SDEs describing finite-particle systems related to the generalised Gaussian ensembles.
PDFXML]]>The generalised sine random point field arises from the scaling limit at the origin of the eigenvalues of the generalised Gaussian ensembles. We solve an infinite-dimensional stochastic differential equation (ISDE) describing an infinite number of interacting Brownian particles which is reversible with respect to the generalised sine random point field. Moreover, finite particle approximation of the ISDE is shown, that is, a solution to the ISDE is approximated by solutions to finite-dimensional SDEs describing finite-particle systems related to the generalised Gaussian ensembles.
PDFXML]]>Yosuke KawamotoMon, 10 Jan 2022 00:00:00 +0200