Optimal insurance demand under marked point processes shocks: a dynamic programming duality approach

TL;DR

This paper develops a duality approach to determine optimal insurance strategies for managing wealth shocks modeled by marked point processes, using stochastic control and viscosity solutions of HJB variational inequalities.

Contribution

It introduces a dual stochastic control framework for optimal insurance under shocks, providing a novel characterization of the value function via viscosity solutions.

Findings

Dual control problem simplifies boundary constraints.

Unique viscosity solution characterizes the value function.

Framework applicable to general marked point process shocks.

Abstract

We study the stochastic control problem of maximizing expected utility from terminal wealth under a non-bankruptcy constraint. The wealth process is subject to shocks produced by a general marked point process. The problem of the agent is to derive the optimal insurance strategy which allows "lowering" the level of the shocks. This optimization problem is related to a suitable dual stochastic control problem in which the delicate boundary constraints disappear. We characterize the dual value function as the unique viscosity solution of the corresponding a Hamilton Jacobi Bellman Variational Inequality (HJBVI in short).