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Optimal investment and risk control problems with delay for an insurer in defaultable market

  • * Corresponding author: Yan Chen

    * Corresponding author: Yan Chen
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  • This paper addresses a investment and risk control problem with a delay for an insurer in the defaultable market. Suppose that an insurer can invest in a risk-free bank account, a risky stock and a defaultable bond. Taking into account the history of the insurer's wealth performance, the controlled wealth process is governed by a stochastic delay differential equation. The insurer's goal is to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. We decompose the original optimization problem into two subproblems: a pre-default case and a post-default case. The explicit solutions in a finite dimensional space are derived for a illustrative situation, and numerical illustrations and sensitivity analysis for our results are provided.

    Mathematics Subject Classification: Primary: 91B30; Secondary: 91B55, 60K05.


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  • Figure 1.  Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal investment strategy $k^{*}(t)$

    Figure 2.  Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal investment strategy $\gamma^{*}(t)$

    Figure 3.  Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal risk control $l^{*}(t)$

    Figure 4.  Value functions with respect to $x$

    Figure 5.  Effect of delay parameters $\beta$ on the pre-default value function

    Figure 6.  Effect of the default parameters $1/\Delta$ and $\zeta$ on the pre-default value function

    Table 1.  Model parameter values

    Symbol Value Symbol Value
    $ \alpha $ $ 0.1 $ $ \nu $ $ 1 $
    $ u $ $ 5 $ $ \lambda $ $ 0.3 $
    $ \beta $ $ 0.3 $ $ \theta $ $ 0.1 $
    $ r $ $ 0.05 $ $ \eta $ $ 0.4 $
    $ \zeta $ $ 0.5 $ $ p $ $ 1 $
    $ \Delta $ $ 0.25 $ $ c $ $ 0.5 $
    $ \mu $ $ 0.15 $ $ \sigma $ $ 0.2 $
     | Show Table
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