Self-Protection and Self-Insurance for General Risk Models via a BSDEs Approach

We investigate an optimal prevention and insurance problem in a general risk setting, where a representative agent is exposed to potential losses. The agent adopts a strategy that combines self-protection, aimed at reducing the frequency of claims, and self-insurance, aimed at mitigating their severity. The problem, which consists in maximizing the expected exponential utility of terminal wealth, is formulated as a stochastic control problem and solved by means of backward stochastic differential equations (BSDEs). Our approach, essentially based on a general Bellman optimality principle (see [ 14 ] among others), does not require specification of the underlying filtration structure, making it applicable to a broad class of risk models, including Markov-modulated, stochastic factor, Cox-shot noise, and self-excited models. We extend recent results by [ 3 , 5 ], which focused on self-protection in specific models, by allowing for both self-protection and self-insurance within a unified and general framework.