# American Institute of Mathematical Sciences

January  2017, 13(1): 375-397. doi: 10.3934/jimo.2016022

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

 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC, 3010, Australia

* Corresponding author: Ping chen

Received  January 2015 Published  March 2016

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.

Citation: Nan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Markowitz's mean-variance optimization with investment and constrained reinsurance. Journal of Industrial & Management Optimization, 2017, 13 (1) : 375-397. doi: 10.3934/jimo.2016022
##### References:

show all references

##### References:
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
The value of $V_\beta (0,X_0)$ in Example 1
The value of $V_\beta (0,X_0)$ in Example 2
Comparisons of efficient frontiers between models with constrained and unconstrained reinsurance
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
 $\max \limits_{\beta \leq \beta_0} V_\beta$ $\max \limits_{\beta_0 \leq \beta \leq \beta_1}\!\!\! V_\beta$ $\max \limits_{\beta_1 \leq \beta \leq \beta_2}\!\!\! V_\beta$ $\max \limits_{\beta \geq \beta_2} V_\beta$ $V_{\beta^*}$ $V_{\beta_0}$ $V_{\hat{\beta}^*}$ $V_{\beta_1}$ $V_{\overline{\beta}^*}$ $V_{\beta_2}$ ${ U(0,1)}$($\!\times\! 1\!0^6$ 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
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
 $\mathop {\max {V_\beta }}\limits_{\beta \leqslant {\beta _0}}$ $\mathop {\max {V_\beta }}\limits_{{\beta _{0 \leqslant }}\beta \leqslant {\beta _1}}$ $\mathop {\max {V_\beta }}\limits_{{\beta _{1 \leqslant }}\beta \leqslant {\beta _2}}$ $\mathop {\max {V_\beta }}\limits_{\beta \geqslant {\beta _2}}$ $V_{\beta^*}$ $V_{\beta_0}$ $V_{\hat{\beta}^*}$ $V_{\beta_1}$ $V_{\overline{\beta}^*}$ $V_{\beta_2}$ ${U(0,1)}$($\times 10^9$) $d\!=\!\frac{d_0+d_2}{2}$ 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
 [1] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 [2] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [3] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [4] Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207 [5] Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 [6] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [7] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [8] Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 [9] Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 [10] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [11] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [12] Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 [13] Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 [14] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [15] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [16] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [17] Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 [18] Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $\Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 1.366