The trapping probability, $ψ$, as defined in Kovacevic and Pflug (2011), is modelled by assuming proportional capital losses, both in the case where there is no insurance and in the case where insurance is purchased by the household. Insurance coverage is likewise proportional, mirroring the structure of quota-share contracts, which are both prevalent in practice and analytically convenient. New closed formulae for $ψ$ are obtained in the case of no insurance when the distribution of the remaining proportion of capital is a power law, extending the results in Kovacevic and Pflug (2011). When proportional insurance is acquired and the remaining proportion of capital is uniformly distributed on $[0,1]$, $ψ$ satisfies a non-local differential equation whose analysis is based on the properties of diffusion processes. The non-local nature of the equation can be addressed using iterative solution methods, leading to a constructive determination of the trapping probability. Constraints on the parameters governing the capital process are derived in both the uninsured and insured cases to prevent the certainty of trapping. Numerical calculations are used to determine the trapping probability for the insured process and to illustrate the impact of different parameters. Consequences on the trapping probability for vulnerable non-poor populations with initial capital slightly above the poverty line are discussed.

翻译：正如Kovacevic和Pflug (2011)所定义的，本文通过假设资本比例损失来建模陷入概率ψ，分别考虑家庭未购买保险和购买保险两种情况。保险保障同样采用比例形式，这与实践中普遍存在且在分析上便于处理的配额共享合同结构相一致。在无保险且资本剩余比例服从幂律分布的情况下，本文推导出ψ的新闭式表达式，拓展了Kovacevic和Pflug (2011)的结果。当购买比例保险且资本剩余比例在[0,1]上均匀分布时，ψ满足一个非局部微分方程，其分析基于扩散过程的性质。该方程的非局部性可通过迭代求解方法处理，从而构造性地确定陷入概率。在无保险和有保险两种情形下，推导出控制资本过程的参数约束条件，以避免陷入的必然性。采用数值计算确定有保险过程下的陷入概率，并展示不同参数的影响。讨论了初始资本略高于贫困线的脆弱非贫困人群的陷入概率后果。