Volume 24, Issue 4

In this paper, Stratonovich curve-line integrals are used to describe the evolution of a stochastic flow driven by some noncommuting vector fields and independent double Wiener processes. In fact, we analyze the corresponding stochastic evolution of a stochastic flow driven by noncommuting vector fields {g1,…,gm}${\{g_{1},\ldots,g_{m}\}}$ and independent double Wiener processes {Wi⁢(t)=(W1i⁢(t1),W2i⁢(t2))∈ℝ2:t=(t1,t2)∈D},1≤i≤m.$\bigl{\{}W^{i}(t)=\bigl{(}W_{1}^{i}(t_{1}),W_{2}^{i}(t_{2})\bigr{)}\in\mathbb{% R}^{2}:t=(t_{1},t_{2})\in D\bigr{\}},\quad 1\leq i\leq m.$ It is a significant generalization of the case m=1${m=1}$, considered in a joint work of V. Damian and C. Vârsan (see [2]). This paper contains two open problems; a good start for a future research.

Wang, Li and Gupta [17] first introduced the skew chi-square distribution based on the multivariate skew normal distribution provided by Azzalini [2], and Ye, Wang and Gupta [18] extended this results into the skew Wishart distribution. Motivated by these results, we first study a new type of multivariate skew normal distribution introduced by Gupta and Chen [12], the moment generating function, independence and quadratic form are discussed, and also a new type of skew chi-square distribution was introduced. Later on, we defined a new type of skew Wishart distribution based on the matrix skew normal models introduced by Ning [15]. In the end, we will study the probabilistic representation of multivariate skew elliptical models.

The article dedicated to twenty-five years of independence of Ukraine and gives a biography of Vyacheslav Girko, written by Stanislav Yezhov in the form of a commented interview.

This paper presents a general form of the covariance matrix structure for a vector random field that is axially symmetric and mean square continuous on the sphere and provides a series representation for a longitudinally reversible one. The series representation is somehow an imitator of the covariance matrix function, and both of them have simpler forms than those proposed in the literature in terms of the associated Legendre functions and are useful for modeling and simulation. Also, a general form of the covariance matrix structure is derived for a spatio-temporal vector random field that is axially symmetric and mean square continuous over the sphere, and a series representation is given for a longitudinally reversible one.

In this paper, we are concerned with a class of fractional partial neutral stochastic integro-differential equations in Hilbert spaces. We assume that the linear part of this equation generates an α-resolvent operator and transform it into an integral equation. By the stochastic analysis and fractional calculus technique, and combining some integral inequalities, we obtain some sufficient conditions ensuring the exponential p -stability of the mild solution of the considered equations are obtained. Subsequently, by the weak convergence approach, we have a try to deal with the stability conditions in distribution of the segment process of mild solutions to the stochastic systems under investigation. Last, an example is presented to illustrate our theory in the work.