# American Institute of Mathematical Sciences

March  2021, 11(1): 1-22. doi: 10.3934/mcrf.2020024

## Optimal dividend policy in an insurance company with contagious arrivals of claims

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

Received  September 2019 Revised  December 2019 Published  March 2020

In this paper we consider the optimal dividend problem for an insurance company whose surplus follows a classical Cramér-Lundberg process with a feature of self-exciting. A Hawkes process is applied so that the occurrence of a jump in the claims triggers more sequent jumps. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition and declare its concavity. We introduce a barrier curve strategy and verify its optimality. Finally, some numerical results are exhibited.

Citation: Yiling Chen, Baojun Bian. Optimal dividend policy in an insurance company with contagious arrivals of claims. Mathematical Control & Related Fields, 2021, 11 (1) : 1-22. doi: 10.3934/mcrf.2020024
##### References:

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##### References:
A sample path of Hawkes process $(N_t,\lambda_t)$ and the surplus process $X_t$ without dividends
Several optimal dividends payment strategy examples
The value function
The fitting barrier curve
The value of $V$ and $V^c$ with $\lambda = 0.5$ and associated barrier points
The barrier curve under different parameter settings (A) the decay rate $\alpha$ (B) the long-run average of the claim intensity $\bar\lambda$ (C) the premium rate $p$ (D) the constant discount factor $c$
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