Modern Stochastics: Theory and Applications

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.

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.

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.

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.

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.

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.

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 $[0,T]$.

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.