# Expected exponential utility maximization of insurers with a general diffusion factor model : The complete market case

## TL;DR

This paper develops a comprehensive model for optimal investment by insurers with exponential utility, considering complex market dynamics influenced by stochastic economic factors, and provides explicit solutions using advanced stochastic control methods.

## Contribution

It introduces a novel framework for insurer investment optimization with nonlinear factor-dependent asset dynamics and solves the associated HJB equation explicitly.

## Key findings

- Derived the HJB equation for the model.
- Proved the unique solvability of the HJB equation.
- Constructed the optimal investment strategy explicitly.

## Abstract

In this paper, we consider the problem of optimal investment by an insurer. The insurer invests in a market consisting of a bank account and $m$ risky assets. The mean returns and volatilities of the risky assets depend nonlinearly on economic factors that are formulated as the solutions of general stochastic differential equations. The wealth of the insurer is described by a Cram\'er--Lundberg process, and the insurer preferences are exponential. Adapting a dynamic programming approach, we derive Hamilton--Jacobi--Bellman (HJB) equation. And, we prove the unique solvability of HJB equation. In addition, the optimal strategy is also obtained using the coupled forward and backward stochastic differential equations (FBSDEs). Finally, proving the verification theorem, we construct the optimal strategy.