The asymptotic ruin problem when the healthy and sick periods form an alternating renewal process

Let {T₁, Y₁}^∞_i=1 be a sequence of positive independent random variables. Let, also, Z₁ = βY′₁ ? πT_i, i = 1, 2, …, where Y′₁ = Max(0, Y_i ? w), w ? 0, and where β < 0 and π is such that E(Z₁) < 0. We consider the random walk of partial sums S_n = ?ⁿ_i=1Z_i in the presence of an absorbing region (u, ∞), u ? 0, and S₀ ≡ 0. Of interest is $\text{ψ(u) = Pr(}\text{S}\text{?}\text{≤ u)}$ where S? = Sup(0, S₁, S₂, …, S_n, …).     

一阶次二次凸Hamilton系统的最小P-边值解

利用对偶变分原理,将一阶次二次凸Hamilton系统的P-边值解问题转化为对偶变分问题,证明对偶泛函满足最小作用原理,进而推导出非平凡P-边值解的存在性结果,最后通过比较作用泛函的临界值,给出该P-边值解的最小周期估计.