Note on the bi-risk discrete time risk model with income rate two

Abstract

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.

Keywords

• Bi-risk model
• discrete time
• finite time survival probability
• ultimate time survival probability
• recursive calculation
• 91G05