The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.
The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.
We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.
In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.
Let $\{L(t),t\ge 0\}$ be a Lévy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic Lévy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process $Z(t)$ is a shifted Lévy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal.
Let $\{L(t),t\ge 0\}$ be a Lévy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic Lévy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process $Z(t)$ is a shifted Lévy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal.
Two optimal transport problems between determinantal point processes (DPP for short) are investigated. It is shown how to estimate the Kantorovitch–Rubinstein and Wasserstein-2 distances between distributions of DPP. These results are applied to evaluate the accuracy of a fast but approximate simulation algorithm of the Ginibre point process restricted to a circle. One can now simulate in a reasonable amount of time more than ten thousands points.