Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Geometric branching reproduction Markov processes, Authors: Assen Tchorbadjieff, Penka Mayster , We present a model of a continuous-time Markov branching process with the infinitesimal generating function defined by the geometric probability distribution. It is proved that the solution of the backward Kolmogorov equation is expressed by the composition of special functions – Wright function in the subcritical case and Lambert-W function in the critical case. We found the explicit form of conditional limit distribution in the subcritical branching reproduction. In the critical case, the extinction probability and probability mass function are expressed as a series containing Bell polynomial, Stirling numbers, and Lah numbers.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation, Authors: Diana Avetisian, Kostiantyn Ralchenko , The paper deals with a stochastic heat equation driven by an additive fractional Brownian space-only noise. We prove that a solution to this equation is a stationary and ergodic Gaussian process. These results enable us to construct a strongly consistent estimator of the diffusion parameter.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: On tail behaviour of stationary second-order Galton–Watson processes with immigration, Authors: Mátyás Barczy, Zsuzsanna Bősze, Gyula Pap , Sufficient conditions are presented on the offspring and immigration distributions of a second-order Galton–Watson process ${({X_{n}})_{n\geqslant -1}}$ with immigration, under which the distribution of the initial values $({X_{0}},{X_{-1}})$ can be uniquely chosen such that the process becomes strongly stationary and the common distribution of ${X_{n}}$, $n\geqslant -1$, is regularly varying.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: On distributions of exponential functionals of the processes with independent increments, Authors: Lioudmila Vostrikova , The aim of this paper is to study the laws of exponential functionals of the processes $X={({X_{s}})_{s\ge 0}}$ with independent increments, namely \[ {I_{t}}={\int _{0}^{t}}\exp (-{X_{s}})ds,\hspace{0.1667em}\hspace{0.1667em}t\ge 0,\] and also \[ {I_{\infty }}={\int _{0}^{\infty }}\exp (-{X_{s}})ds.\] Under suitable conditions, the integro-differential equations for the density of ${I_{t}}$ and ${I_{\infty }}$ are derived. Sufficient conditions are derived for the existence of a smooth density of the laws of these functionals with respect to the Lebesgue measure. In the particular case of Lévy processes these equations can be simplified and, in a number of cases, solved explicitly.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: On infinite divisibility of a class of two-dimensional vectors in the second Wiener chaos, Authors: Andreas Basse-O’Connor, Jan Pedersen, Victor Rohde , Infinite divisibility of a class of two-dimensional vectors with components in the second Wiener chaos is studied. Necessary and sufficient conditions for infinite divisibility are presented as well as more easily verifiable sufficient conditions. The case where both components consist of a sum of two Gaussian squares is treated in more depth, and it is conjectured that such vectors are infinitely divisible.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Approximations of the ruin probability in a discrete time risk model, Authors: David J. Santana, Luis Rincón , Based on a discrete version of the Pollaczeck–Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Simple approximations for the ruin probability in the risk model with stochastic premiums and a constant dividend strategy, Authors: Olena Ragulina , We deal with a generalization of the risk model with stochastic premiums where dividends are paid according to a constant dividend strategy and consider heuristic approximations for the ruin probability. To be more precise, we construct five- and three-moment analogues to the De Vylder approximation. To this end, we obtain an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes. Finally, we analyze the accuracy of the approximations for some typical distributions of premium and claim sizes using statistical estimates obtained by the Monte Carlo methods.

Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Distance from fractional Brownian motion with associated Hurst index 0, Authors: Oksana Banna, Filipp Buryak, Yuliya Mishura , We find the best approximation of the fractional Brownian motion with the Hurst index $H\in (0,1/2)$ by Gaussian martingales of the form ${\textstyle\int _{0}^{t}}{s^{\gamma }}d{W_{s}}$, where W is a Wiener process, $\gamma >0$.