Modern Stochastics: Theory and Applications

Journal Information
ISSN / EISSN : 2351-6046 / 2351-6054
Published by: VTeX (10.15559)
Total articles ≅ 237
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Published: 13 September 2022
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Modern Stochastics: Theory and Applications pp 1-18; https://doi.org/10.15559/22-vmsta214

Abstract:
Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $\check{\text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.
Published: 6 September 2022
by VTeX
Modern Stochastics: Theory and Applications pp 483-498; https://doi.org/10.15559/22-vmsta213

Abstract:
A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as $t\to \infty $ is studied.
Published: 2 September 2022
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Modern Stochastics: Theory and Applications pp 453-482; https://doi.org/10.15559/22-vmsta212

Abstract:
In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.
Published: 5 July 2022
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Modern Stochastics: Theory and Applications pp 431-452; https://doi.org/10.15559/22-vmsta211

Abstract:
In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three parameters gives us a freedom to operate with the processes and we get a wider application possibilities. On the other hand, it leads to the need to apply rather subtle methods, depending on the intervals where the parameters fall. Integration with respect to the processes under consideration is defined, and it is found for which parameters the processes are differentiable. Finally, the Volterra representation is inverted, that is, the representation of the underlying Wiener process via Gaussian Volterra process is found. Therefore, it is shown that for any indices for which Gaussian Volterra process is defined, it generates the same flow of sigma-fields as the underlying Wiener process – the property that has been used many times when considering a fractional Brownian motion.
Published: 21 June 2022
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Modern Stochastics: Theory and Applications pp 413-430; https://doi.org/10.15559/22-vmsta210

Abstract:
The need to model a Markov renewal on-off process with multiple off-states arise in many applications such as economics, physics, and engineering. Characterization of the occupation time of one specific off-state marginally or two off-states jointly is crucial to understand such processes. The exact marginal and joint distributions of the off-state occupation times are derived. The theoretical results are confirmed numerically in a simulation study. A special case when all holding times have Lévy distribution is considered for the possibility of simplification of the formulas.
Published: 20 June 2022
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Modern Stochastics: Theory and Applications pp 401-412; https://doi.org/10.15559/22-vmsta209

Abstract:
This article provides survival probability calculation formulas for bi-risk discrete time risk model with income rate two. More precisely, the possibility for the stochastic process $u+2t-{\textstyle\sum _{i=1}^{t}}{X_{i}}-{\textstyle\sum _{j=1}^{\lfloor t/2\rfloor }}{Y_{j}}$, $u\in \mathbb{N}\cup \{0\}$, to stay positive for all $t\in \{1,\hspace{0.1667em}2,\hspace{0.1667em}\dots ,\hspace{0.1667em}T\}$, when $T\in \mathbb{N}$ or $T\to \infty $, is considered, where the subtracted random part consists of the sum of random variables, which occur in time in the following order: ${X_{1}},\hspace{0.1667em}{X_{2}}+{Y_{1}},\hspace{0.1667em}{X_{3}},\hspace{0.1667em}{X_{4}}+{Y_{2}},\hspace{0.1667em}\dots $ Here ${X_{i}},\hspace{0.1667em}i\in \mathbb{N}$, and ${Y_{j}},\hspace{0.1667em}j\in \mathbb{N}$, are independent copies of two independent, but not necessarily identically distributed, nonnegative and integer-valued random variables X and Y. Following the known survival probability formulas of the similar bi-seasonal model with income rate two, $u+2t-{\textstyle\sum _{i=1}^{t}}{X_{i}}{\mathbb{1}_{\{i\hspace{2.5pt}\text{is odd}\}}}-{\textstyle\sum _{j=1}^{t}}{Y_{i}}{\mathbb{1}_{\{j\hspace{2.5pt}\text{is even}\}}}$, it is demonstrated how the bi-seasonal model is used to express survival probability calculation formulas in the bi-risk case. Several numerical examples are given where the derived theoretical statements are applied.
Published: 14 June 2022
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Modern Stochastics: Theory and Applications pp 377-399; https://doi.org/10.15559/22-vmsta208

Abstract:
In mixture with varying concentrations model (MVC) one deals with a nonhomogeneous sample which consists of subjects belonging to a fixed number of different populations (mixture components). The population which a subject belongs to is unknown, but the probabilities to belong to a given component are known and vary from observation to observation. The distribution of subjects’ observed features depends on the component which it belongs to. Generalized estimating equations (GEE) for Euclidean parameters in MVC models are considered. Under suitable assumptions the obtained estimators are asymptotically normal. A jackknife (JK) technique for the estimation of their asymptotic covariance matrices is described. Consistency of JK-estimators is demonstrated. An application to a model of mixture of nonlinear regressions and a real life example are presented.
Published: 7 June 2022
by VTeX
Modern Stochastics: Theory and Applications pp 339-355; https://doi.org/10.15559/22-vmsta207

Abstract:
The existence of the bifractional Brownian motion ${B_{H,K}}$ indexed by a sphere when $K\in (-\infty ,1]\setminus \{0\}$ and $H\in (0,1/2]$ is discussed, and the asymptotics of its excursion probability $\mathbb{P}\left\{{\sup _{M\in \mathbb{S}}}{B_{H,K}}(M)>x\right\}$ as $x\to \infty $ is studied.
Published: 7 June 2022
by VTeX
Modern Stochastics: Theory and Applications pp 357-375; https://doi.org/10.15559/22-vmsta206

Abstract:
A new, direct proof of the formulas for the conic intrinsic volumes of the Weyl chambers of types ${A_{n-1}}$, ${B_{n}}$ and ${D_{n}}$ is given. These formulas express the conic intrinsic volumes in terms of the Stirling numbers of the first kind and their B- and D-analogues. The proof involves an explicit determination of the internal and external angles of the faces of the Weyl chambers.
Published: 27 April 2022
by VTeX
Modern Stochastics: Theory and Applications pp 313-338; https://doi.org/10.15559/22-vmsta205

Abstract:
The stochastic process of the form \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] 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 $[0,T]$.
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