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:
[1]

S. AsmussenB. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075.  Google Scholar

[2]

L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar

[4]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[5]

P. ChenH. L. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[6]

T. ChoulliM. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative Finance, 1 (2001), 573-596.  doi: 10.1088/1469-7688/1/6/301.  Google Scholar

[7]

T. ChoulliM. Taksar and X. Y. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.  doi: 10.1137/S0363012900382667.  Google Scholar

[8]

M. G. Crandell and P. Lions, Viscosity solution of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[9]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[10]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998 (1998), 166-180.  doi: 10.1016/S0167-6687(98)00007-9.  Google Scholar

[11]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models with transaction costs, Insurance: Mathematics and Economics, 22 (1998), 41-51.  doi: 10.1016/S0167-6687(98)00007-9.  Google Scholar

[12]

B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar

[13]

C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004.  Google Scholar

[14]

X. LiX. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 41 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[15]

P. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Ⅱ. Viscosity solutions and uniqueness, Communications in Partial Differential Equations, 8 (1983), 1229-1276.  doi: 10.1080/03605308308820301.  Google Scholar

[16]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[17]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68.  doi: 10.1080/034612301750077338.  Google Scholar

[18]

H. Schmidli, On minimising the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.  Google Scholar

[19]

S. Z. Shi, Convex Analysis, Shanghai Science and Technology Press, 1990. Google Scholar

[20]

H. M. Soner, Optimal control with state-space constrain Ⅰ, SIAM Journal on Control and Optimization, 24 (1986), 552-561.  doi: 10.1137/0324032.  Google Scholar

[21]

H. M. Soner, Optimal control with state-space constrain Ⅱ, SIAM Journal on Control and Optimization, 24 (1986), 1110-1122.  doi: 10.1137/0324067.  Google Scholar

[22]

M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar

[23]

Z. W. WangJ. M. Xia and L. H. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003.  Google Scholar

[24]

H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[25]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[26]

X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance: Mathematics and Economics, 45 (2009), 81-88.  doi: 10.1016/j.insmatheco.2009.04.001.  Google Scholar

[27]

X. ZhangM. Zhou and J. Y. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Applied Stochastic Models in Business and Industry, 23 (2007), 63-71.  doi: 10.1002/asmb.637.  Google Scholar

[28]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

show all references

References:
[1]

S. AsmussenB. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075.  Google Scholar

[2]

L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.  Google Scholar

[4]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[5]

P. ChenH. L. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[6]

T. ChoulliM. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative Finance, 1 (2001), 573-596.  doi: 10.1088/1469-7688/1/6/301.  Google Scholar

[7]

T. ChoulliM. Taksar and X. Y. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.  doi: 10.1137/S0363012900382667.  Google Scholar

[8]

M. G. Crandell and P. Lions, Viscosity solution of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[9]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[10]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998 (1998), 166-180.  doi: 10.1016/S0167-6687(98)00007-9.  Google Scholar

[11]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models with transaction costs, Insurance: Mathematics and Economics, 22 (1998), 41-51.  doi: 10.1016/S0167-6687(98)00007-9.  Google Scholar

[12]

B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar

[13]

C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004.  Google Scholar

[14]

X. LiX. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 41 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[15]

P. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Ⅱ. Viscosity solutions and uniqueness, Communications in Partial Differential Equations, 8 (1983), 1229-1276.  doi: 10.1080/03605308308820301.  Google Scholar

[16]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[17]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68.  doi: 10.1080/034612301750077338.  Google Scholar

[18]

H. Schmidli, On minimising the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.  Google Scholar

[19]

S. Z. Shi, Convex Analysis, Shanghai Science and Technology Press, 1990. Google Scholar

[20]

H. M. Soner, Optimal control with state-space constrain Ⅰ, SIAM Journal on Control and Optimization, 24 (1986), 552-561.  doi: 10.1137/0324032.  Google Scholar

[21]

H. M. Soner, Optimal control with state-space constrain Ⅱ, SIAM Journal on Control and Optimization, 24 (1986), 1110-1122.  doi: 10.1137/0324067.  Google Scholar

[22]

M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar

[23]

Z. W. WangJ. M. Xia and L. H. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003.  Google Scholar

[24]

H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[25]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[26]

X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance: Mathematics and Economics, 45 (2009), 81-88.  doi: 10.1016/j.insmatheco.2009.04.001.  Google Scholar

[27]

X. ZhangM. Zhou and J. Y. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Applied Stochastic Models in Business and Industry, 23 (2007), 63-71.  doi: 10.1002/asmb.637.  Google Scholar

[28]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

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
$\max \limits_{\beta \leq \beta_0} V_\beta$ $\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
$\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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