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General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes

  • * Corresponding author: Ran Xu

    * Corresponding author: Ran Xu
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  • For spectrally negative Lévy risk processes we consider a generalized version of the De Finetti's optimal dividend problem with fixed transaction costs, where the ruin time is replaced by a general drawdown time in the framework. We identify a condition under which a band–type impulse dividend strategy is optimal among all admissible impulse strategies. As a consequence, we are able to extend the previous results on ruin time based impulse dividend optimization problem to those on drawdown time based impulse dividend optimization problems. A new type of drawdown function is proposed at end, and various numerical examples are presented to illustrate the existence of those optimal impulse dividend strategies under different assumptions.

    Mathematics Subject Classification: Primary: 60G51, 91G50; Secondary: 60E10.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  Illustration of various drawdown times

    Figure 2.  General drawdown times based on Eq.(43)

    Figure 3.  Cramér-Lundberg model: $ \varsigma(z) $

    Figure 4.  Cramér-Lundberg model: $ V^{z_2^*}_{z_1^*}(x) $

    Figure 5.  Brownian Motion with drift: $ \varsigma(z) $

    Figure 6.  Brownian Motion with drift: $ V^{z_2^*}_{z_1^*}(x) $

    Figure 7.  Jump-diffusion process: $ \varsigma(z) $

    Figure 8.  Jump-diffusion process: $ V^{z_2^*}_{z_1^*}(x) $

    Table 1.  Cramér-Lundberg model

    $ (z_1^*, z_2^*) $ $ k= \infty $ $ k=0.6 $ $ k=0.4 $ $ k=0.2 $
    $ \alpha=2.0 $ (3.9689, 11.6898) (3.9689, 11.6902) (2.0000, 11.2002) (2.0000, 10.1503)
    $ \alpha=1.5 $ (3.4688, 11.1898) (3.4688, 11.1893) (1.5000, 10.0581) (1.5330, 10.3051)
    $ \alpha=1.2 $ (3.1688, 10.8898) (3.1680, 10.8917) (3.1688, 10.8898) (1.6154, 10.3557)
    $ \alpha=1.0 $ (2.9688, 10.6898) (2.9687, 10.6901) (2.9688, 10.6894) (1.7244, 10.3693)
     | Show Table
    DownLoad: CSV

    Table 2.  Cramér-Lundberg model: $ (z_1^*, z_2^*) $ vs. $ \beta $

    $ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.0, k=0.2 $ Ruin Time
    $ \beta=0.5 $ (4.5855, 10.5305) (4.0855, 10.0305) (2.6070, 9.3218) (2.5855, 8.5305)
    $ \beta=1.0 $ (3.9689, 11.6898) (3.4688, 11.1893) (1.7244, 10.3693) (1.9688, 9.6898)
    $ \beta=1.5 $ (3.5369, 12.6025) (3.0369, 12.1025) (1.0605, 11.1901) (1.5369, 10.6025)
    $ \beta=2.0 $ (3.1911, 13.3968) (1.5000, 12.8618) (1.0000, 11.9132) (1.1912, 11.3968)
    $ \beta= 2.5 $ (2.8968, 14.1194) (1.5000, 13.5114) (1.0000, 12.6039) (0.8968, 12.1194)
     | Show Table
    DownLoad: CSV

    Table 3.  Brownian Motion with drift

    $ (z_1^*, z_2^*) $$ k = \infty $$ k = 0.6 $$ k = 0.4 $$ k = 0.2 $
    $ \alpha = 2.0 $(3.8442, 12.3162)(3.8441, 12.3176)(3.0519, 11.7522)(2.2989, 10.8580)
    $ \alpha = 1.5 $(3.3442, 11.8162)(3.3441, 11.8162)(3.0575, 11.6979)(2.2989, 10.8580)
    $ \alpha = 1.2 $(3.0442, 11.5162)(3.0441, 11.5162)(3.0397, 11.5094)(2.2989, 10.8580)
    $ \alpha = 1.0 $(2.8442, 11.3162)(2.8439, 11.3162)(2.8441, 11.3166)(2.2989, 10.8580)
     | Show Table
    DownLoad: CSV

    Table 4.  Brownian motion with drift: $ (z_1^*, z_2^*) $ vs. $ \beta $

    $ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.0, k=0.2 $ Ruin Time
    $ \beta=0.5 $ (3.9695, 9.9869) (3.4695, 9.4869) (2.4539, 8.5596) (1.9695, 7.9869)
    $ \beta=1.0 $ (3.8442, 12.3162) (3.3441, 11.8162) (2.2989, 10.8580) (1.8442, 10.3162)
    $ \beta=1.5 $ (3.7664, 14.1668) (3.2664, 13.6668) (2.2024, 12.6891) (1.7664, 12.1668)
    $ \beta=2.0 $ (3.7087, 15.7657) (3.2087, 15.2657) (2.1307, 14.2736) (1.7087, 13.7657)
    $ \beta= 2.5 $ (3.6623, 17.2027) (3.1623, 16.7027) (2.0730, 15.6991) (1.6623, 15.2027)
     | Show Table
    DownLoad: CSV

    Table 5.  Jump-diffusion process

    $ \alpha=2.0 $ (5.4394, 14.7877) (5.4394, 14.7879) (5.4393, 14.7878) (2.0000, 13.4730)
    $ \alpha=1.5 $ (4.9394, 14.2876) (4.9394, 14.2876) (4.9394, 14.2875) (2.1303, 13.7094)
    $ \alpha=1.2 $ (4.6394, 13.9876) (4.6394, 13.9876) (4.6386, 13.9915) (2.7529, 13.7678)
    $ \alpha=1.0 $ (4.4394, 13.7876) (4.4394, 13.7865) (4.4394, 13.7877) (3.5068, 13.7285)
     | Show Table
    DownLoad: CSV

    Table 6.  Jump-diffusion process: $ (z_1^*, z_2^*) $ vs. $ \beta $

    $ (z_1^*, z_2^*) $ $ \alpha=2.0, k=\infty $ $ \alpha=1.5, k=0.6 $ $ \alpha=1.2, k=0.4 $ Ruin Time
    $ \beta=0.5 $ (6.2530, 13.5120) (5.7530, 13.0120) (5.2514, 12.5135) (4.2530, 11.5120)
    $ \beta=1.0 $ (5.4394, 14.7876) (4.9394, 14.2876) (3.5068, 13.7285) (3.4394, 12.7876)
    $ \beta=1.5 $ (4.8685, 15.7668) (4.3685, 15.2668) (2.0611, 14.5761) (2.8685, 13.7667)
    $ \beta=2.0 $ (4.4172, 16.6037) (3.9172, 16.1037) (1.2175, 15.3164) (2.4172, 14.6037)
    $ \beta= 2.5 $ (4.0462, 17.3555) (3.5462, 16.8555) (1.0000, 15.9652) (2.0462, 15.3555)
     | Show Table
    DownLoad: CSV
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