Self-protection and self-insurance for general risk models via a BSDE approach

TL;DR

This paper develops a BSDE-based framework for optimal prevention and insurance strategies in general risk models, enabling broad applicability without specific filtration assumptions.

Contribution

It introduces a unified approach to optimize self-protection and self-insurance in diverse risk settings using BSDEs, extending previous model-specific results.

Findings

Framework applicable to various risk models

Optimal strategies derived for combined self-protection and insurance

Generalized approach without filtration restrictions

Abstract

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 [13] 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.