Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model

This paper considers the problem of optimal investment-reinsurance for the insurer and reinsurer under the constant elasticity of variance (CEV) model. It is assumed that the net claims process is approximated by a diffusion process, both the insurer and reinsurer can invest in risk-free assets and risky assets. We use the variance premium principle to calculate the premiums of the insurer and reinsurer, and the reinsurance proportion is constrained by the net profit condition. Our objective is to maximize the joint exponential utility of the insurer and reinsurer's terminal wealth for a fixed time. By solving the HJB equation, we obtain the explicit expressions of the optimal investment-reinsurance strategy and value function. We find that the optimal reinsurance strategy can be divided into many cases and is related to the risk aversion coefficient of the insurer and reinsurer, but independent of the price of risky assets. Furthermore, we give the proof of the verification theorem. Finally, we demonstrate a numerical analysis to explain the results

Additional file 1 of The hepato-ovarian axis: genetic evidence for a causal association between non-alcoholic fatty liver disease and polycystic ovary syndrome

Additional file 1: Table S1. Key characteristics of participating studies. Table S2. GWAS significant SNPs used as genetic instruments for fasting insulin and fasting glucose. Table S3. GWAS significant SNPs used as genetic instruments for serum SHBG levels and bioavailable testosterone levels in women. Table S4. Direct causal effects of NAFLD, fasting insulin, fasting glucose, serum SHBG levels, and serum bioavailable testosterone levels on PCOS risk via multivariable MR analysis. Table S5. Direct causal effects of NAFLD, fasting insulin, fasting glucose, and serum SHBG levels on serum bioavailable testosterone levels via multivariable MR analysis. Table S6. Direct causal effects of NAFLD, fasting insulin, and fasting glucose on serum SHBG levels via multivariable MR analysis. Table S7. Obesity-related genome-wide significant genetic variants. Table S8. Directional pleiotropy test using MR-Egger intercepts. Table S9. Horizontal pleiotropy test using MR-PRESSO. Table S10. Linkage disequilibrium score regression results on genetic correlations between NAFLD, fasting insulin, fasting glucose, SHBG, BT, and PCOS. Table S11. Indirect causal effects between NAFLD and PCOS via fasting insulin, serum SHBG levels, and serum bioavailable testosterone levels through step-wise MR analysis