In this paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Levy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

Consider a compound renewal risk model, in which a single accident may cause more than one claim. Under the condition that the common distribution of the individual claims is second order subexponential, we establish a second order asymptotic formula for the infinite-time ruin probability. Compared with the traditional ones, our second order asymptotic result is more precise and effective, which can be demonstrated by the numerical studies.

It has been observed in real networks that the fraction of nodes P(k) with degree k satisfies the power-law P(k) ∝ k−γ for k > kmin > 0. However, the degree distribution of nodes in these networks before kmin varies slowly to the extent of being uniform as compared to the degree distribution after kmin. Most of the previous studies focus on the degree distribution after kmin and ignore the initial flatness in the distribution of degrees. In this paper, we propose a model that describes the degree distribution for the whole range of k > 0, i.e., before and after kmin. The network evolution is made up of two steps. In the first step, a new node is connected to the network through a preferential attachment method. In the second step, a certain number of edges between the existing nodes are added such that the end nodes of an edge are selected either uniformly or preferentially. The model has a parameter to control the uniform or preferential selection of nodes for creating edges in the network. We perform a comprehensive mathematical analysis of our proposed model in the discrete domain and prove that the model exhibits an asymptotically power-law degree distribution after kmin and a flat-ish distribution before kmin. We also develop an algorithm that guides us in determining the model parameters in order to fit the model output to the node degree distribution of a given real network. Our simulation results show that the degree distributions of the graphs generated by this model match well with those of the real-world graphs.

A matrix variate generalization of the two-sided power distribution is proposed. Several properties of this distribution including probability density function, cumulative distribution function and characteristic function are derived.

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GIX/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GIX/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GIX/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

Association or interdependence of two stock prices is analyzed, and selection criteria for a suitable model developed in the present paper. The association is generated by stochastic correlation, given by a stochastic differential equation (SDE), creating interdependent Wiener processes. These, in turn, drive the SDEs in the Heston model for stock prices. To choose from possible stochastic correlation models, two goodness-of-fit procedures are proposed based on the copula of Wiener increments. One uses the confidence domain for the centered Kendall function, and the other relies on strong and weak tail dependence. The constant correlation model and two different stochastic correlation models, given by Jacobi and hyperbolic tangent transformation of Ornstein-Uhlenbeck (HtanOU) processes, are compared by analyzing daily close prices for Apple and Microsoft stocks. The constant correlation, i.e., the Gaussian copula model, is unanimously rejected by the methods, but all other two are acceptable at a 95% confidence level. The analysis also reveals that even for Wiener processes, stochastic correlation can create tail dependence, unlike constant correlation, which results in multivariate normal distributions and hence zero tail dependence. Hence models with stochastic correlation are suitable to describe more dangerous situations in terms of correlation risk.

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

In this paper we show how to extend a simple common shock model with Archimedean dependence of the hidden variables to the non-exchangeable case. The assumption is that the hidden risk factors are linked by a hierarchical Archimedean dependence structure, possibly fully nested. We give directions about how to implement the model and to address the issue that the hidden variables must be put in descending dependence order. We show how the model can be simplified in the Gumbel-Marshall-Olkin distribution in Cherubini and Mulinacci (2017), the only case in which exponential distribution of the observed variables is preserved.

In this paper, we propose proximal splitting-type algorithms for sampling from distributions whose densities are not necessarily smooth nor log-concave. Our approach brings together tools from, on the one hand, variational analysis and non-smooth optimization, and on the other hand, stochastic diffusion equations, and in particular the Langevin diffusion. We establish in particular consistency guarantees of our algorithms seen as discretization schemes in this context. These algorithms are then applied to compute the exponentially weighted aggregates for regression problems involving non-smooth penalties that are commonly used to promote some notion of simplicity/complexity. Some popular penalties are detailed and implemented on some numerical experiments.