Optimal Dividend of Compound Poisson Process under a Stochastic Interest Rate

TL;DR

This paper investigates optimal dividend strategies for an insurance company with a wealth process driven by a compound Poisson process, considering stochastic interest rates modeled by geometric Brownian motion and Ornstein-Uhlenbeck processes, providing explicit solutions and properties.

Contribution

It derives explicit optimal dividend strategies under geometric Brownian motion interest rates and characterizes the value function as a viscosity solution in the Ornstein-Uhlenbeck case.

Findings

Explicit optimal strategies for geometric Brownian motion interest rates.

Properties of the value function under Vasicek model.

Value function as viscosity solution in complex cases.

Abstract

In this paper we assume the insurance wealth process is driven by the compound Poisson process. The discounting factor is modelled as a geometric Brownian motion at first and then as an exponential function of an integrated Ornstein-Uhlenbeck process. The objective is to maximize the cumulated value of expected discounted dividends up to the time of ruin. We give an explicit expression of the value function and the optimal strategy in the case of interest rate following a geometric Brownian motion. For the case of the Vasicek model, we explore some properties of the value function. Since we can not find an explicit expression for the value function in the second case, we prove that the value function is the viscosity solution of the corresponding HJB equation.