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# Markowitz's mean-variance optimization with investment and constrained reinsurance

• * Corresponding author: Ping chen
• This paper deals with the optimal investment-reinsurance strategy for an insurer under the criterion of mean-variance. The risk process is the diffusion approximation of a compound Poisson process and the insurer can invest its wealth into a financial market consisting of one risk-free asset and one risky asset, while short-selling of the risky asset is prohibited. On the side of reinsurance, we require that the proportion of insurer's retained risk belong to $[0, 1]$, is adopted. According to the dynamic programming in stochastic optimal control, the resulting Hamilton-Jacobi-Bellman (HJB) equation may not admit a classical solution. In this paper, we construct a viscosity solution for the HJB equation, and based on this solution we find closed form expressions of efficient strategy and efficient frontier when the expected terminal wealth is greater than a certain level. For other possible expected returns, we apply numerical methods to analyse the efficient frontier. Several numerical examples and comparisons between models with constrained and unconstrained proportional reinsurance are provided to illustrate our results.

Mathematics Subject Classification: Primary: 91G10, 91G80; Secondary: 70H20.

 Citation: • • Figure 1.  the region of $\mathcal{A}_i,\: i=1,2,3,4.$ For the region $\mathcal{A}_4$, we can deem it as a family of curves $\{ \mathcal{C}_k \} _{0\leq k\leq 1}$ (i.e., the red dot curve) and construct a solution to the HJB equation on each curve

Figure 2.  The value of $V_\beta (0,X_0)$ in Example 1

Figure 3.  The value of $V_\beta (0,X_0)$ in Example 2

Figure 4.  Comparisons of efficient frontiers between models with constrained and unconstrained reinsurance

Table 1.  Piecewise and global maximum values of $V_\beta (0,X_0)$ under different distributions, if $\lambda=10$, $\theta=0.3$, $\eta=0.2$, $\mu=0.06$, $r=0.04$, $\sigma=1$, $T=100$ and $X_0=50$, which lead to $d_1 < d_2$ in all the following distributions

 $\max \limits_{\beta \leq \beta_0} V_\beta$ $\max \limits_{\beta_0 \leq \beta \leq \beta_1}\!\!\! V_\beta$ $\max \limits_{\beta_1 \leq \beta \leq \beta_2}\!\!\! V_\beta$ $\max \limits_{\beta \geq \beta_2} V_\beta$ $V_{\beta^*}$ $V_{\beta_0}$ $V_{\hat{\beta}^*}$ $V_{\beta_1}$ $V_{\overline{\beta}^*}$ $V_{\beta_2}$ ${ U(0,1)}$($\!\times\! 1\!0^6$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -4.1533 -0.0037 -0.0037 $\underline {{\mathbf{1}}{\mathbf{.7}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{26}}}}}$ -0.0037 $d\!=\!\frac{d_1+d_2}{2}$ N/A -0.9965 $\underline {{\mathbf{6}}{\mathbf{.5318}}}$ -0.8962 N/A -1.1406 $d\!=\!\frac{d_2+\overline{d}}{2} \$ $\underline {{\mathbf{0}}{\mathbf{.2173}}}$ 0.1306 0.1306 -3.8089 N/A -4.2972 ${Exp(1)}$($\!\times \!1\!0^7$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -1.6809 -0.0033 -0.0033 $\underline {{\mathbf{9}}{\mathbf{.1}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{16}}}}}$ -0.0033 $d\!=\!\frac{d_1+d_2}{2}$ N/A -0.3927 $\underline {{\mathbf{77}}{\mathbf{.814}}}$ -0.3368 N/A -0.4836 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{0}}{\mathbf{.1596}}$ 0.0816 0.0816 -1.4784 N/A -1.7716 ${\Gamma(2,1)}$($\!\times\! 1\!0^7$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -6.6657 -0.0075 -0.0075 $\underline {{\mathbf{6}}{\mathbf{.3}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{22}}}}}$ -0.0075 $d\!=\!\frac{d_1+d_2}{2}$ N/A -1.5892 $\underline {{\mathbf{144}}{\mathbf{.61}}}$ -1.4119 N/A -1.8521 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{0}}{\mathbf{.4130}}$ 0.2376 0.2376 -6.0489 N/A -6.9279 ${Erlang(3,\!0.5)}$($\!\times \!1\!0^8$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -5.9556 -0.0037 -0.0037 $\underline {{\mathbf{2}}{\mathbf{.3}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{30}}}}}$ -0.0037 $d\!=\!\frac{d_1+d_2}{2}$ N/A -1.4403 $\underline {{\mathbf{633}}{\mathbf{.53}}}$ -1.3172 N/A -1.6104 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{0}}{\mathbf{.2413}}$ 0.1546 0.1546 -5.5396 N/A -6.1254 ${Pareto(3,1)}$($\!\times \!1\!0^6$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -4.3176 -0.0331 -0.0331 $\underline {{\mathbf{5}}{\mathbf{.4}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{6}}}}}$ -0.0331 $d\!=\!\frac{d_1+d_2}{2}$ N/A -0.9050 $\underline {{\mathbf{245}}{\mathbf{.60}}}$ -0.6891 N/A -1.4259 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{1}}{\mathbf{.2362}}$ 0.4564 -3.3675 -3.3675 N/A -4.8359 ${N(1,2^2)}$($\!\times\! 1\!0^7$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -1.7430 -0.0207 -0.0207 $\underline {{\mathbf{0}}{\mathbf{.0030}}}$ -0.0207 $d\!=\!\frac{d_1+d_2}{2}$ N/A -0.3394 $\underline {{\mathbf{2611}}{\mathbf{.7}}}$ -0.2474 N/A -0.6164 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{0}}{\mathbf{.7276}}$ 0.2403 0.2403 -1.2836 N/A -2.0184 ${LN(1,1)}$($\!\times\! 1\!0^8$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -3.4138 -0.0123 -0.0123 $\underline {{\mathbf{4}}{\mathbf{.9}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{9}}}}}$ -0.0123 $d\!=\!\frac{d_1+d_2}{2}$ N/A -0.7710 $\underline {{\mathbf{4187}}{\mathbf{.9}}}$ -0.6309 N/A -1.0322 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{0}}{\mathbf{.5216}}$ 0.2323 0.2323 -2.8733 N/A -3.6740 ${NB(1,0.6)}$($\!\times\! 1\!0^7$ except underline) $d\!=\!\frac{d_0+d_1}{2}$ N/A -3.8213 -0.0132 -0.0132 $\underline {{\mathbf{2}}{\mathbf{.8}}{\mathbf{ \times }}{\mathbf{1}}{{\mathbf{0}}^{{\mathbf{ - }}{\mathbf{10}}}}}$ -0.0132 $d\!=\!\frac{d_1+d_2}{2}$ N/A -0.8653 $\underline {{\mathbf{438}}{\mathbf{.12}}}$ -0.7103 N/A -1.1514 $d\!=\!\frac{d_2+\overline{d}}{2} \$ ${\mathbf{0}}{\mathbf{.5664}}$ 0.2545 0.2545 -3.2263 N/A -4.1063

Table 2.  Piecewise and global maximum values of $V_\beta (0,X_0)$ under different distributions, if $\lambda=1$, $\theta=0.25$, $\eta=0.2$, $\mu=0.12$, $r=0.1$, $\sigma=1$, $T=100$ and $X_0=50$, which lead to $d_1 > d_2$ in all the following distributions

 $\mathop {\max {V_\beta }}\limits_{\beta \leqslant {\beta _0}}$ $\mathop {\max {V_\beta }}\limits_{{\beta _{0 \leqslant }}\beta \leqslant {\beta _1}}$ $\mathop {\max {V_\beta }}\limits_{{\beta _{1 \leqslant }}\beta \leqslant {\beta _2}}$ $\mathop {\max {V_\beta }}\limits_{\beta \geqslant {\beta _2}}$ $V_{\beta^*}$ $V_{\beta_0}$ $V_{\hat{\beta}^*}$ $V_{\beta_1}$ $V_{\overline{\beta}^*}$ $V_{\beta_2}$ ${U(0,1)}$($\times 10^9$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -0.9978 -0.9212 -1.8896 ${\mathbf{0}}{\mathbf{.0020}}$ -0.2225 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 0.8971 ${\mathbf{0}}{\mathbf{.9296}}$ -0.8106 0.0080 -0.8842 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{7}}{\mathbf{.4450}}$ 2.3088 2.3166 -0.1859 0.0173 -1.9383 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{49}}{\mathbf{.763}}$ 3.8965 3.8965 -0.1519 N/A -5.2191 ${Exp(1)}$($\times 10^{10}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -0.5612 -0.5220 -1.9780 ${\mathbf{0}}{\mathbf{.0042}}$ -0.0960 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 0.5807 ${\mathbf{0}}{\mathbf{.6015}}$ -1.1763 0.0169 -0.3828 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{9}}{\mathbf{.3795}}$ 2.1070 2.1098 -0.2391 0.0587 -1.3311 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{51}}{\mathbf{.957}}$ 3.3527 3.3527 0.0744 N/A -4.0010 ${\Gamma(2,1)}$($\times 10^{10}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -1.7584 -1.6274 -4.0605 ${\mathbf{0}}{\mathbf{.0055}}$ -4.0605 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 1.6593 ${\mathbf{1}}{\mathbf{.7189}}$ -1.9674 0.0219 -1.4494 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{17}}{\mathbf{.205}}$ 4.7277 4.7395 -0.4418 0.0548 -3.6266 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{106}}{\mathbf{.69}}$ 7.7884 7.7884 -0.1778 N/A -10.075 ${Erlang(3,\!0.5)}$($\times 10^{11}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -1.2426 -1.2423 -1.6745 ${\mathbf{0}}{\mathbf{.0011}}$ -0.3123 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 1.0630 ${\mathbf{1}}{\mathbf{.1002}}$ -0.5359 0.0043 -1.2483 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{5}}{\mathbf{.6680}}$ 2.2385 2.2512 -0.1273 0.0077 -2.2031 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{44}}{\mathbf{.398}}$ 3.9729 3.9727 -0.2576 N/A -5.6482 ${Pareto(3,1)}$($\times 10^{10}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -0.2622 -0.2459 -1.9025 ${\mathbf{0}}{\mathbf{.0075}}$ -0.0297 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 0.3471 ${\text{0}}{\text{.3584}}$ -1.3874 0.0297 -0.1179 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{9}}{\mathbf{.0697}}$ 1.8675 1.8690 -0.1602 0.1934 -0.7671 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{51}}{\mathbf{.540}}$ 3.1176 3.1176 0.6238 N/A -2.9664 ${N(1,2^2)}$($\times 10^{11}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -0.1294 -0.1215 -1.1009 ${\mathbf{0}}{\mathbf{.0050}}$ -0.0131 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 0.1890 ${\mathbf{0}}{\mathbf{.1948}}$ -0.8222 0.0198 -0.0522 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{4}}{\mathbf{.9701}}$ 1.0826 1.0836 -0.0647 0.1456 -0.3834 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{30}}{\mathbf{.540}}$ 1.9036 1.9036 0.5233 N/A -1.6600 ${LN(1,1)}$($\times 10^{11}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -1.4785 -1.3810 -7.5183 ${\mathbf{0}}{\mathbf{.0223}}$ -0.2086 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 1.7070 ${\mathbf{1}}{\mathbf{.7662}}$ -5.0451 0.0890 -0.8339 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{37}}{\mathbf{.143}}$ 7.5727 7.5794 -0.8606 0.4226 -3.9611 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{198}}{\mathbf{.97}}$ 12.037 12.037 1.0593 N/A -13.110 ${NB(1,0.6)}$($\times 10^{10}$) $d\!=\!\frac{d_0+d_2}{2}$ N/A -1.6279 -1.5202 -8.1096 ${\mathbf{0}}{\mathbf{.0236}}$ -0.2324 $d\!=\!\!d_3\!\!-\!\!1\!00$ N/A 1.8624 ${\mathbf{1}}{\mathbf{.9273}}$ -5.4140 0.0942 -0.9276 $d\!=\!\frac{d_1+d_2}{2}$ ${\mathbf{40}}{\mathbf{.041}}$ 8.1885 8.1958 -0.9357 0.4402 -4.3335 $d\!=\!\frac{d_1+\overline{d}}{2}$ ${\mathbf{214}}{\mathbf{.47}}$ 13.003 13.003 1.0824 N/A -14.249
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