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

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  • 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.

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

    Citation:

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  • Figure 1.  A sample path of Hawkes process $ (N_t,\lambda_t) $ and the surplus process $ X_t $ without dividends

    Figure 2.  Several optimal dividends payment strategy examples

    Figure 3.  The value function

    Figure 4.  The fitting barrier curve

    Figure 5.  The value of $ V $ and $ V^c $ with $ \lambda = 0.5 $ and associated barrier points

    Figure 6.  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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