Optimal stopping of a risk process when claims are covered immediately

TL;DR

This paper studies an optimal stopping problem for an insurance risk process with immediate claim payments, investment, inflation, and rejection options, providing a new model and solution approach.

Contribution

It introduces a novel risk process model incorporating immediate claims payment schemes and rejection options, extending existing models and solutions.

Findings

Derived a new optimal stopping strategy for the enhanced risk process.

Showed the impact of immediate claims payment on the company's capital.

Provided analytical solutions for the modified risk process.

Abstract

The optimal stopping problem for the risk process with interests rates and when claims are covered immediately is considered. An insurance company receives premiums and pays out claims which have occured according to a renewal process and which have been recognized by them. The capital of the company is invested at interest rate $\alpha\in\Re^{+}$, the size of claims increase at rate $\beta\in\Re^{+}$ according to inflation process. The immediate payment of claims decreases the company investment by rate $\alpha_1$. The aim is to find the stopping time which maximizes the capital of the company. The improvement to the known models by taking into account different scheme of claims payment and the possibility of rejection of the request by the insurance company is made. It leads to essentially new risk process and the solution of optimal stopping problem is different.