Volume 23, Issue 1

In this paper, we present a random variable to which the Bernoulli type Galton–Watson branching process with immigration converges in law. This convergence is based on the limiting distribution of the process given in [Commun. Stoch. Anal. 5 (2011), no. 3, 457–480]. We also give another proof of the result in [Commun. Stoch. Anal. 5 (2011), no. 3, 457–480] in this paper.

In this article, we evaluate the integral ∫01∫01xα+g-1yα+h-1(1-x)β+k-1(1-y)β+ℓ-1|x-y|2γdydx,$ \int _{0}^{1}\int _{0}^{1} x^{\alpha +g-1}y^{\alpha +h -1}(1-x)^{\beta +k-1} (1-y)^{\beta +\ell -1} |{x -y}|^{2\gamma }\,\mathrm {d}y\,\mathrm {d}x , $ where α > 0, β > 0, γ ≥ 0, and g , h , k , ℓ are positive integers. Explicit expressions of the above integral are derived in simple elementary functions for various values of g , h , k , and ℓ.

Azzalini [Scand. J. Stat. 12 (1985), 171–178] first introduced the skew normal distribution family with a shape parameter λ, and then extended this family by adding an additional shape parameters ξ. Basic properties of these two families were studied. Henze [Scand. J. Stat. 13 (1986), 271–275] gave the probabilistic representations for these two families by interpreting it as the linear combination of a normal random variable with another normal random variable truncated at the origin and several properties were illustrated. Chen and Gupta [Statistics 39 (2005), no. 3, 247–253] extended the skew normal distribution family to the matrix variate and proposed the moment generating function and the quadratic form of the matrix variate skew normal models. Motivated by these results, we first study the probabilistic representation for the matrix variate skew normal models and several properties. Then we define the extended skew normal model of the matrix variate, and give the probabilistic representation for this family and its extension.

In this paper we will consider the periodic AGARCH( p , q ) (Periodic Asymmetric Generalized Autoregressive Conditional Heteroscedastic) process, denoted PAGARCH( p , q ). These processes are similar to the standard AGARCH processes but now include seasonally varying coefficients. We examine the probabilistic structure of a PAGARCH-type stochastic difference equation with periodically varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions basing on the top Lyapunov exponent. Furthermore, we examine the necessary and sufficient conditions ensuring the existence of a unique causal periodically correlated (PC) solution to our processes.

This paper is concerned with the optimal control problem for stochastic differential delay equations with Poisson jumps, in which both state and control delays are involved. We obtain the necessary and sufficient maximum principles for the optimal control by virtue of the duality method and the anticipated backward stochastic differential equations with Poisson jumps. An optimal consumption rate problem is discussed to illustrate the applications of our results.

We present a complete study of the ergodic theorem for the problem of nonlinear Klein–Gordon classical wave propagations through cylindrical measures, rigorous mathematical path integrals and the well-known Ruelle–Amrein–Georgescu–Enss (RAGE) theorem on the characterization of the continuous spectrum of self-adjoint operators.