optimal investment and dividend policy in an insurance company: A varied bound for dividend rates

Yiling Chen; Baojun Bian

doi:10.3934/dcdsb.2019044

Article Contents

2019, Volume 24, Issue 9: 5083-5105. Doi: 10.3934/dcdsb.2019044

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$ ^{4} $

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optimal investment and dividend policy in an insurance company: A varied bound for dividend rates

Yiling Chen and
Baojun Bian

School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Received: June 30, 2018

Revised: September 30, 2018

Early access: February 2019

Published: July 2019

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Abstract

In this paper we consider an optimal dividend problem for an insurance company whose surplus process evolves a classical $ {\rm Cram\acute{e}r} $-Lundberg process. We impose a varied bound over the dividend rate to raise the dividend payment at a acceptable survival probability. Our objective is to find a strategy consisting of both investment and dividend payment which maximizes the cumulative expected discounted dividend payment until the ruin time. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition. We characterize the optimal value function as the smallest viscosity supersolution of the HJB equation. We introduce a method to construct the potential solution of our problem and give a verification theorem to check its optimality. Finally we show some numerical results.

Keywords:

Insurance,
optimal dividend payment and investment strategy,
varied bound,
viscosity solution,
band strategy.

Mathematics Subject Classification: Primary: 35J87, 91B30, 49J20; Secondary: 49L25, 91B70, 49K20.

Citation:

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Figure 1. (a) The optimal value function. (b) The optimal dividend payment strategy. (c) The the optimal investment policy. (d) The first order derivative of $ V(x) $

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Figure 2. Optimal value function for $ p = 4 $ and different $ g $ restrictions

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Figure 3. Survival probability function under optimal strategy for $ p = 2 $ and different $ g $ restrictions

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