Optimal investment problem in a renewal risk model with generalized Erlang distributed interarrival times

TL;DR

This paper investigates the optimal investment strategy in a renewal risk model with generalized Erlang interarrival times, deriving explicit solutions for the strategy and value function under different interest rate conditions.

Contribution

It provides explicit formulas for optimal investment policies in a renewal risk model with generalized Erlang interarrival times, including cases with zero and nonzero interest rates.

Findings

Explicit optimal investment strategies derived

Concavity of the value function established

Solutions applicable under different interest rate scenarios

Abstract

This paper explores the optimal investment problem of a renewal risk model with generalized Erlang distributed interarrival times. The phases of the Erlang interarrival time is assumed to be observable. The price of the risky asset is driven by the constant elasticity of variance model (CEV) and the insurer aims to maximize the exponential utility of the terminal wealth by asset allocation. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we establish the concavity of the value function and derive an explicit expression for the optimal investment policy when the interest rate is zero. When the interest rate is nonzero, we obtain an explicit form of the optimal investment strategy, along with a semi-explicit expression of the value function, whose concavity is also rigorously proven.