齐次离散时间风险模型中源于生存概率的$\mathbf{2\times2}$决定因素
On $\mathbf{2\times2}$ determinants originating from survival probabilities in homogeneous discrete time risk model
摘要：
分析了齐次离散时间风险模型$W(n)=u+κn+\sum_{i=1}^nZ_i$中最终时间生存概率$φ(u)$的初值问题中出现的$2\times 2$ Hankel型行列式$D_n$，其中$Z_i$为正整数值i.i.d.随机索赔，初始盈余$u \in \mathbb{N}_0$和收入率$κ=2$。我们证明了最近的一个猜想的非零和单调性$D_n$的渐近版本，并推导出明确的公式的初始值$φ(0)$，$φ(1)$的复发，产生生存概率。当$Z_i$服从Bernoulli分布或几何分布时，关于$D_n$的猜想对所有$n\in\mathbb{N}_0$都成立.此外，生成函数$Ξ(s)$的最终生存概率$φ(u)$的推导。
英文摘要：
We analyze $2\times 2$ Hankel-like determinants $D_n$ that arise in the initial values problem for the ultimate time survival probability $φ(u)$ in a homogeneous discrete time risk model $W(n)=u+κn+\sum_{i=1}^nZ_i$, where $Z_i$ are positive integer valued i.i.d. random claims, the initial surplus $u \in \mathbb{N}_0$ and the income rate $κ=2$. We prove the asymptotic version of a recent conjecture on the non--vanishing and monotonicity of $D_n$ and derive explicit formulas for the initial values $φ(0)$, $φ(1)$ of a recurrence that yields survival probabilities. In cases when $Z_i$ are Bernoulli or Geometrically distributed, the conjecture on $D_n$ is shown to hold for all $n\in\mathbb{N}_0$. Additionally, a generating function $Ξ(s)$ for ultimate survival probabilities $φ(u)$ is derived.
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