On the impact of insurance on households susceptible to random proportional losses: An analysis of poverty trapping

In this paper, we consider a risk process with deterministic growth and multiplicative jumps to model the capital of a low-income household. Reflecting the high-risk nature of the low-income environment, capital losses are assumed to be proportional to the level of accumulated capital at the jump time. Our aim is to derive the probability that a household falls below the poverty line, i.e. the trapping probability, where ``trapping" occurs when the level of capital of a household holds falls below the poverty line, to an area from which it is difficult to escape without external help. Considering the remaining proportion of capital to be distributed as a special case of the beta distribution, closed-form expressions for the trapping probability are obtained via analysis of the Laplace transform of the infinitesimal generator of the process. To study the impact of insurance on this probability, introduction of an insurance product offering proportional coverage is presented. The infinitesimal generator of the insured process gives rise to non-local differential equations. To overcome this, we propose a recursive method for deriving a closed-form solution of the integro-differential equation associated with the infinitesimal generator of the insured process and provide a numerical estimation method for obtaining the trapping probability. Constraints on the rate parameters of the process that prevent certain trapping are derived in both the uninsured and insured cases using classical results from risk theory.