Modern Stochastics: Theory and Applications

In this paper, the distribution function and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.

This article provides survival probability calculation formulas for bi-risk discrete time risk model with income rate two. More precisely, the possibility for the stochastic process $u+2t-{\textstyle\sum _{i=1}^{t}}{X_{i}}-{\textstyle\sum _{j=1}^{\lfloor t/2\rfloor }}{Y_{j}}$, $u\in \mathbb{N}\cup \{0\}$, to stay positive for all $t\in \{1,\hspace{0.1667em}2,\hspace{0.1667em}\dots ,\hspace{0.1667em}T\}$, when $T\in \mathbb{N}$ or $T\to \infty $, is considered, where the subtracted random part consists of the sum of random variables, which occur in time in the following order: ${X_{1}},\hspace{0.1667em}{X_{2}}+{Y_{1}},\hspace{0.1667em}{X_{3}},\hspace{0.1667em}{X_{4}}+{Y_{2}},\hspace{0.1667em}\dots $ Here ${X_{i}},\hspace{0.1667em}i\in \mathbb{N}$, and ${Y_{j}},\hspace{0.1667em}j\in \mathbb{N}$, are independent copies of two independent, but not necessarily identically distributed, nonnegative and integer-valued random variables X and Y. Following the known survival probability formulas of the similar bi-seasonal model with income rate two, $u+2t-{\textstyle\sum _{i=1}^{t}}{X_{i}}{\mathbb{1}_{\{i\hspace{2.5pt}\text{is odd}\}}}-{\textstyle\sum _{j=1}^{t}}{Y_{i}}{\mathbb{1}_{\{j\hspace{2.5pt}\text{is even}\}}}$, it is demonstrated how the bi-seasonal model is used to express survival probability calculation formulas in the bi-risk case. Several numerical examples are given where the derived theoretical statements are applied.

This paper deals with the discrete-time risk model with nonidentically distributed claims. We suppose that the claims repeat with time periods of three units, that is, claim distributions coincide at times $\{1,4,7,\dots \}$, at times $\{2,5,8,\dots \}$, and at times $\{3,6,9,\dots \}$. We present the recursive formulas to calculate the finite-time and ultimate ruin probabilities. We illustrate the theoretical results by several numerical examples.