[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Equilibrium reinsurance-investment strategies with partial information and common shock dependence

  • Original Research
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In this paper, we study an optimal reinsurance-investment problem with partial information and common shock dependence under the mean-variance criterion for an insurer. The insurer has two dependent classes of insurance business, which are subject to a common shock. We consider the optimal reinsurance-investment problem under complete information and partial information, respectively. We formulate the complete information problem within a game theoretic framework and seek the equilibrium reinsurance-investment strategy and equilibrium value function by solving an extended Hamilton–Jacobi–Bellman system of equations. For the partial information problem, we first transform it to a completely observable model by virtue of the filtering theory, then derive the equilibrium strategy and equilibrium value function by using the methods similar to those for the complete information problem. In addition, we illustrate the equilibrium reinsurance-investment strategies by numerical examples and discuss the impacts of model parameters on the equilibrium reinsurance-investment strategies for both the complete information and partial information cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. A pre-committed strategy means that if the decision makers can commit themselves at the initial time, they can choose a strategy that is optimal from the point of view at the initial time, and then constrain themselves to abide by it in the future, although the strategy is no longer optimal for the future time.

References

  • Asadi, M., Ebrahimi, N., & Soofi, E. S. (2018). Optimal hazard models based on partial information. European Journal of Operational Research, 270(2), 723–733.

    Google Scholar 

  • Bai, L., Cai, J., & Zhou, M. (2013). Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting. Insurance Mathematics and Economics, 53(3), 664–670.

    Google Scholar 

  • Bäuerle, N. (2005). Benchmark and mean-variance problems for insurers. Mathematical Methods of Operations Research, 62(1), 159–165.

    Google Scholar 

  • Ben Abdelaziz, F. (2012). Solution approaches for the multiobjective stochastic programming. European Journal of Operational Research, 216(1), 1–16.

    Google Scholar 

  • Ben Abdelaziz, F., Aouni, B., & El Fayedh, R. (2007). Multi-objective stochastic programming for portfolio selection. European Journal of Operational Research, 177(3), 1811–1823.

    Google Scholar 

  • Ben Abdelaziz, F., & Masri, H. (2010). A compromise solution for the multiobjective stochastic linear programming under partial uncertainty. European Journal of Operational Research, 202(1), 55–59.

    Google Scholar 

  • Ben Abdelaziz, F., Masri, H., & Alaya, H. (2017). A recourse goal programming approach for airport bus routing problem. Annals of Operations Research, 251(1–2), 383–396.

    Google Scholar 

  • Bi, J., & Cai, J. (2019). Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets. Insurance Mathematics and Economics, 85, 1–14.

    Google Scholar 

  • Bi, J., Jin, H., & Meng, Q. (2018). Behavioral mean-variance portfolio selection. European Journal of Operational Research, 271(2), 644–663.

    Google Scholar 

  • Bi, J., Liang, Z., & Xu, F. (2016). Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence. Insurance Mathematics and Economics, 70, 245–258.

    Google Scholar 

  • Bielecki, T. R., Jin, H., Pliska, S. R., & Zhou, X. Y. (2005). Continuous-time mean-variance portfolio selection with bankruptcy probihition. Mathematical Finance, 15, 213–244.

    Google Scholar 

  • Björk, T., Davis, M. H. A., & Landén, C. (2010). Optimal investment under partial information. Mathematical Methods of Operations Research, 71, 371–399.

    Google Scholar 

  • Björk, T., Khapko, M., & Murgoci, A. (2017). On time-inconsistent stochastic control in continuous time. Finance and Stochastics, 21, 331–360.

    Google Scholar 

  • Björk, T., & Murgoci, A. (2010). A general theory of Markovian time inconsistent stochastic control problems. Working paper. Available at: http://ssrn.com/abstract=1694759.

  • Björk, T., & Murgoci, A. (2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics, 18, 545–592.

    Google Scholar 

  • Björk, T., Murgoci, A., & Zhou, X. Y. (2014). Mean-variance portfolio optimization with state-dependent risk aversion. Mathematical Finance, 24(1), 1–24.

    Google Scholar 

  • Brendle, S. (2006). Portfolio selection under incomplete information. Stochastic Processes and their Applications, 116, 701–723.

    Google Scholar 

  • Browne, S. (1995). Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Mathematics of Operations Research, 20(4), 937–958.

    Google Scholar 

  • Cong, J. F., & Tan, K. S. (2016). Optimal VaR-based risk management with reinsurance. Annals of Operations Research, 237(1–2), 177–202.

    Google Scholar 

  • Deng, C., Zeng, X. D., & Zhu, H. M. (2018). Non-zero-sum stochastic differential reinsurance and investment games with default risk. European Journal of Operational Research, 264(3), 1144–1158.

    Google Scholar 

  • Dimitrova, D. S., Kaishev, V. K., & Zhao, S. Q. (2015). On finite-time ruin probabilities in a generalized dual risk model with dependence. European Journal of Operational Research, 242(1), 134–148.

    Google Scholar 

  • Feng, B., Zhao, J., & Jiang, Z. (2021). Robust pricing for airlines with partial information. Annals of Operations Research. https://doi.org/10.1007/s10479-020-03926-9

    Article  Google Scholar 

  • Fouque, J. P., Papanicolaou, A., & Sircar, A. (2015). Filtering and portfolio optimization with stochastic unobserved drift in asset returns. Communications in Mathematical Sciences, 13(4), 935–953.

    Google Scholar 

  • Gu, A., Viens, F. G., & Yi, B. (2017). Optimal reinsurance and investment strategies for insurers with mispricing and model ambiguity. Insurance Mathematics and Economics, 72, 235–249.

    Google Scholar 

  • Jang, B., Kim, K. T., & Lee, H. (2021). Optimal reinsurance and portfolio selection: Comparison between partial and complete information models. European Financial Management. https://doi.org/10.1111/eufm.12303

    Article  Google Scholar 

  • Kacem, I., La Torre, D., & Masri, H. (2021). Preface: Recent advances in multiple objective optimization and goal programming. Annals of Operations Research, 296(1–2), 1–5.

    Google Scholar 

  • Kim, J. H., Kim, W. C., Kwon, D. G., & Fabozzi, F. J. (2018). Robust equity portfolio performance. Annals of Operations Research, 266(1–2), 293–312.

    Google Scholar 

  • La Torre, D., & Mendivil, F. (2018). Portfolio optimization under partial uncertainty and incomplete information: A probability multimeasure-based approach. Annals of Operations Research, 267(1–2), 267–279.

    Google Scholar 

  • Li, Y., & Li, Z. (2013). Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion. Insurance Mathematics and Economics, 53(1), 86–97.

    Google Scholar 

  • Li, D., & Ng, W. L. (2000). Optimal dynamic portfolio selection: Multiperiod mean-variance formulation. Mathematical Finance, 10(3), 387–406.

    Google Scholar 

  • Li, Y., Qiao, H., Wang, S., & Zhang, L. (2015). Time-consistent investment strategy under partial information. Insurance Mathematics and Economics, 65, 187–197.

    Google Scholar 

  • Li, S., & Qiu, Z. (2021). Equilibrium investment-reinsurance strategy with delay and common shock dependence under Heston’s SV model. Optimization. https://doi.org/10.1080/02331934.2021.1935934

  • Li, D., Shen, Y., & Zeng, Y. (2018). Dynamic derivative-based investment strategy for mean-variance asset-liability management with stochastic volatility. Insurance Mathematics and Economics, 78, 72–86.

    Google Scholar 

  • Li, Y., Wang, S., Zeng, Y., & Qiao, H. (2017). Equilibrium investment strategy for a DC plan with partial information and mean-variance criterion. IEEE Systems Journal, 11(3), 1492–1504.

    Google Scholar 

  • Li, X., Zhou, X. Y., & Lim, A. E. B. (2002). Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM Journal on Control and Optimization, 40, 1540–1555.

    Google Scholar 

  • Liang, Z., & Song, M. (2015). Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information. Insurance Mathematics and Economics, 65, 66–76.

    Google Scholar 

  • Liang, Z., & Yuen, K. C. (2016). Optimal dynamic reinsurance with dependent risks: Variance premium principle. Scandinavian Actuarial Journal, 2016(1), 18–36.

    Google Scholar 

  • Liang, Z., Yuen, K. C., & Zhang, C. (2018). Optimal reinsurance and investment in a jump-diffusion financial market with common shock dependence. Journal of Applied Mathematics and Computing, 56, 637–664.

    Google Scholar 

  • Liptser, R., & Shiryayev, A. (2004). Statistics of Random Processes, I, II. Berlin: Springer.

    Google Scholar 

  • Liu, J., & Chen, Z. (2018). Time consistent multi-period robust risk measures and portfolio selection models with regime-switching. European Journal of Operational Research, 268(1), 373–385.

    Google Scholar 

  • Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.

    Google Scholar 

  • Masri, H. (2017). A multiple stochastic goal programming approach for the agent portfolio selection problem. Annals of Operations Research, 251(1–2), 179–192.

    Google Scholar 

  • Masri, H. (2018). A Shariah-compliant portfolio selection model. Journal of the Operational Research Society, 69(10), 1568–1575.

    Google Scholar 

  • Peng, X., & Hu, Y. (2013). Optimal proportional reinsurance and investment under partial information. Insurance Mathematics and Economics, 53(2), 416–428.

    Google Scholar 

  • Promislow, D., & Young, V. (2005). Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal, 9(3), 110–128.

    Google Scholar 

  • Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability, 12(3), 890–907.

    Google Scholar 

  • Wang, L., Chen, Z., & Yang, P. (2021). Robust equilibrium control-measure policy for a DC pension plan with state-dependent risk aversion under mean-variance criterion. Journal of Industrial and Management Optimization, 17(3), 1203–1233.

    Google Scholar 

  • Wei, J., & Wang, T. (2017). Time-consistent mean-variance asset-liability management with random coefficients. Insurance Mathematics and Economics, 77, 84–96.

    Google Scholar 

  • Wu, H., & Zeng, Y. (2015). Equilibrium investment strategy for defined-contribution pension schemes with generalized mean-variance criterion and mortality risk. Insurance Mathematics and Economics, 64, 396–408.

    Google Scholar 

  • Xu, L., Zhang, L., & Yao, D. (2017). Optimal investment and reinsurance for an insurer under Markov-modulated financial market. Insurance Mathematics and Economics, 74, 7–19.

    Google Scholar 

  • Yuen, K. C., Guo, J., & Wu, X. (2006). On the first time of ruin in the bivariate compound Poisson model. Insurance Mathematics and Economics, 38, 298–308.

    Google Scholar 

  • Yuen, K. C., Liang, Z., & Zhou, M. (2015). Optimal proportional reinsurance with common shock dependence. Insurance Mathematics and Economics, 64, 1–13.

    Google Scholar 

  • Zeng, Y., & Li, Z. (2011). Optimal time-consistent investment and reinsurance policies for mean-variance insurers. Insurance Mathematics and Economics, 49(1), 145–154.

    Google Scholar 

  • Zeng, Y., Li, D., & Gu, A. (2016). Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps. Insurance Mathematics and Economics, 66, 138–152.

    Google Scholar 

  • Zhang, C., & Liang, Z. (2017). Portfolio optimization for jump-diffusion risky assets with common shock dependence and state dependent risk aversion. Optimal Control Applications and Methods, 38(2), 229–246.

    Google Scholar 

  • Zhou, X. Y., & Li, D. (2000). Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42, 19–33.

    Google Scholar 

  • Zhou, Z., Ren, T., Xiao, H., & Liu, W. (2019). Time-consistent investment and reinsurance strategies for insurers under multi-period mean-variance formulation with generalized correlated returns. Journal of Management Science and Engineering, 4(2), 142–157.

    Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous reviewer and the Editor for the insightful comments and suggestions that improved the presentation of the paper. Junna Bi is supported by National Natural Science Foundation of China (Nos. 11871220, 11871219, 12071224) and 111 Project (No. B14019). Jun Cai acknowledges the financial support from the Natural Sciences and Engineering Research Council of Canada (No. RGPIN-2016-03975). Yan Zeng is supported by National Natural Science Foundation of China (Nos. 71991474, 71771220, 71571195, 71721001).

Author information

Authors and Affiliations

  1. School of Statistics, KLATASDS-MOE, East China Normal University, Shanghai, 200241, People’s Republic of China

    Junna Bi

  2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, N2L 3G1, Canada

    Jun Cai

  3. Lingnan (University) College, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Yan Zeng

Authors
  1. Junna Bi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yan Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan Zeng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: The proof of Theorem 2

Before giving the proof of Theorem 2, we first provide Proposition 1 and Lemma 1 as follows.

Proposition 1

The extended HJB system of Eq. (3.4) can be simplified as follows

$$\begin{aligned}&(V_{c})_{t}+ \mathop {\sup }_{\pi \in {\varPi }_{c}}\bigg \{\left[ rx+(\mu -r)u+c_{1}q_{1}+c_{2}q_{2}+c_{3}\right] (V_{c})_{x}+\kappa (\delta -\mu )(V_{c})_{\mu } +\frac{1}{2}\sigma _{1}^{2} u^{2}(V_{c})_{xx}\nonumber \\&\qquad +\frac{1}{2}(\sigma _{12}^{2}+\sigma _{2}^{2})(V_{c})_{\mu \mu }+\sigma _{1}\sigma _{12}u (V_{c})_{x\mu }-\frac{\gamma }{2}\sigma _{1}^{2} u^{2}[(g_{c})_{x}]^{2}-\frac{\gamma }{2}(\sigma _{12}^{2}+\sigma _{2}^{2})[(g_{c})_{\mu }]^{2}\nonumber \\&\qquad -\gamma \sigma _{1}\sigma _{12}u(g_{c})_{x}(g_{c})_{\mu }\nonumber \\&\qquad +\lambda _{1}{\mathbb {E}}\left[ V_{c}(t,x-q_{1}L,\mu )-V_{c}(t,x,\mu )-\frac{\gamma }{2}(g_{c}(t,x-q_{1}L,\mu )-g_{c}(t,x,\mu ) )^{2} \right] \nonumber \\&\qquad +\lambda _{2}{\mathbb {E}}\left[ V_{c}(t,x-q_{2}Y,\mu )-V_{c}(t,x,\mu )-\frac{\gamma }{2}(g_{c}(t,x-q_{2}Y,\mu )-g_{c}(t,x,\mu ) )^{2} \right] \nonumber \\&\qquad +\lambda {\mathbb {E}}\left[ V_{c}(t,x-q_{1}L-q_{2}Y,\mu )-V_{c}(t,x,\mu )-\frac{\gamma }{2}(g_{c}(t,x-q_{1}L-q_{2}Y,\mu )-g_{c}(t,x,\mu ) )^{2} \right] \bigg \} \nonumber \\&\quad =0, \nonumber \\&V_{c}(T,x,\mu )=x, \end{aligned}$$
(7.1)
$$\begin{aligned}&(g_{c})_{t} + \left[ rx+(\mu -r)u^{*}+c_{1}q_{1}^{*}+c_{2}q_{2}^{*}+c_{3}\right] (g_{c})_{x}+\kappa (\delta -\mu )(g_{c})_{\mu }\nonumber \\&\qquad +\frac{1}{2}\sigma _{1}^{2} {u^{*}}^{2}(g_{c})_{xx}+\frac{1}{2}(\sigma _{12}^{2}+\sigma _{2}^{2})(g_{c})_{\mu \mu } \nonumber \\&\qquad +\sigma _{1}\sigma _{12}u^{*} (g_{c})_{x\mu }+\lambda _{1}{\mathbb {E}}\left[ g_{c}(t,x-q_{1}^{*}L,\mu )-g_{c}(t,x,\mu )\right] \nonumber \\&\qquad +\lambda _{2}{\mathbb {E}}\left[ g_{c}(t,x-q_{2}^{*}Y,\mu )-g_{c}(t,x,\mu ) \right] \nonumber \\&\qquad +\lambda {\mathbb {E}}\left[ g_{c}(t,x-q_{1}^{*}L-q_{2}^{*}Y,\mu )-g_{c}(t,x,\mu ) \right] =0, \nonumber \\&g_{c}(T,x,\mu )=x. \end{aligned}$$
(7.2)

Proof

According to the model assumptions, after a straightforward but tedious calculation, we find that the extended HJB system of Eq. (3.4) has the simplified form (7.1–7.2). We omit the detailed calculation. \(\square \)

Lemma 1

For two-dimensional function \(f(x,y)={\tilde{a}}x^{2}+{\tilde{b}}y^{2}+{\tilde{c}}x+{\tilde{d}}y+{\tilde{e}}xy\), if \({\tilde{a}}>0\) and \(4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}>0\), f(x, y) attains its minimum value \(f(x^{*},y^{*})=-\frac{{\tilde{b}}{\tilde{c}}^{2}+{\tilde{a}}{\tilde{d}}^{2}-{\tilde{c}}{\tilde{d}}{\tilde{e}}}{4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}}\) at \((x^{*},y^{*})=\left( \frac{{\tilde{e}}{\tilde{d}}-2{\tilde{b}}{\tilde{c}}}{4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}}, \frac{{\tilde{e}}{\tilde{c}}-2{\tilde{a}}{\tilde{d}}}{4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}}\right) .\)

Proof

The lemma follows the standard solutions for an extreme value problem of a two-dimensional function. The detailed arguments are omitted. \(\square \)

The proof of Theorem 2

Since the wealth process has the linear structure, and in accordance with the forms of the boundary conditions, we conjecture that

$$\begin{aligned}&V_{c}(t,x,\mu )=A(t)x+B(t)\mu ^{2}+C(t)\mu +D(t),\\&g_{c}(t,x,\mu )=a(t)x+b(t)\mu ^{2}+c(t)\mu +d(t). \end{aligned}$$

Then, we have

$$\begin{aligned}&(V_{c})_{t}=A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t},\\&(V_{c})_{x}=A,\ (V_{c})_{xx}=0,\ (V_{c})_{\mu }=2B\mu +C, \ (V_{c})_{\mu \mu }=2B, \ (V_{c})_{x\mu }=0, \end{aligned}$$

and

$$\begin{aligned}&(g_{c})_{t}=a_{t}x+b_{t}\mu ^{2}+c_{t}\mu +d_{t},\\&(g_{c})_{x}=a, \ (g_{c})_{xx}=0,\ (g_{c})_{\mu }=2b\mu +c, \ (g_{c})_{\mu \mu }=2b, \ (g_{c})_{x\mu }=0. \end{aligned}$$

Substituting the above expressions into (7.1) and (7.2), we have

$$\begin{aligned}&A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t}+ \mathop {\sup }_{\pi \in {\varPi }_{c}}\bigg \{\left[ rx+(\mu -r)u+c_{1}q_{1}+c_{2}q_{2}+c_{3}\right] A+\kappa (\delta -\mu )(2B\mu +C) \nonumber \\&\quad +(\sigma _{12}^{2}+\sigma _{2}^{2})B-\frac{\gamma }{2}\sigma _{1}^{2} a^{2}u^{2}-\frac{\gamma }{2}(\sigma _{12}^{2}+\sigma _{2}^{2})(2b\mu +c)^{2}-\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)u\nonumber \\&\quad +\lambda _{1}\left( -A q_{1}\mu _{1L}-\frac{\gamma }{2}a^{2}q_{1}^{2}\mu _{2L} \right) +\lambda _{2}\left( -A q_{2}\mu _{1Y}-\frac{\gamma }{2}a^{2}q_{2}^{2}\mu _{2Y} \right) \nonumber \\&\quad +\lambda \left[ -A q_{1}\mu _{1L}-A q_{2}\mu _{1Y}-\frac{\gamma }{2}a^{2}(q_{1}^{2}\mu _{2L}+q_{2}^{2}\mu _{2Y}+2q_{1}q_{2}\mu _{1L}\mu _{1Y}) \right] \bigg \}=0, \nonumber \\&A(T)=1,~B(T)=C(T)=D(T)=0, \end{aligned}$$
(7.3)

and

$$\begin{aligned}&a_{t}x+b_{t}\mu ^{2}+c_{t}\mu +d_{t}+ \left[ rx+(\mu -r)u^{*}+c_{1}q_{1}^{*}+c_{2}q_{2}^{*}+c_{3}\right] a+\kappa (\delta -\mu )(2b\mu +c), \nonumber \\&\quad +(\sigma _{12}^{2}+\sigma _{2}^{2})b+\lambda _{1}(-aq_{1}^{*}\mu _{1L})+\lambda _{2}(-aq_{2}^{*}\mu _{1Y})+\lambda (-aq_{1}^{*}\mu _{1L}-aq_{2}^{*}\mu _{1Y})=0, \nonumber \\&a(T)=1,~b(T)=c(T)=d(T)=0. \end{aligned}$$
(7.4)

Rearrange (7.3), we have

$$\begin{aligned}&A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t}+ \mathop {\sup }_{\pi \in {\varPi }_{c}}\bigg \{-\frac{\gamma }{2}\sigma _{1}^{2} a^{2}u^{2}+\left[ (\mu -r)A-\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)\right] u -f(q_{1},q_{2}) \nonumber \\&\quad +\left[ rx+c_{3}\right] A+\kappa (\delta -\mu )(2B\mu +C)+(\sigma _{12}^{2}+\sigma _{2}^{2})B-\frac{\gamma }{2}(\sigma _{12}^{2} +\sigma _{2}^{2})(2b\mu +c)^{2} \bigg \}=0, \end{aligned}$$
(7.5)

where

$$\begin{aligned} f(q_{1},q_{2}) =&\frac{\gamma }{2}(\lambda +\lambda _{1})a^{2}\mu _{2L}q_{1}^{2}+\frac{\gamma }{2}(\lambda +\lambda _{2})a^{2}\mu _{2Y}q_{2}^{2} -\eta _{1}(\lambda +\lambda _{1})\mu _{1L}Aq_{1}\\&-\eta _{2}(\lambda +\lambda _{2})\mu _{1Y}Aq_{2}+\lambda \gamma a^{2} \mu _{1L}\mu _{1Y}q_{1}q_{2}. \end{aligned}$$

From (7.5), we have

$$\begin{aligned} u^{*}&=\frac{(\mu -r)A-\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)}{\gamma \sigma _{1}^{2}a^{2}}. \end{aligned}$$
(7.6)

Next we look for the equilibrium reinsurance strategy and the corresponding equilibrium value function. We firstly give the detailed discussion for Case 2.

For Case 2, using Lemma 1, we obtain the equilibrium reinsurance strategy

$$\begin{aligned}&q_{1}^{*}=\frac{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\eta _{1}\mu _{1L}\mu _{2Y}-\lambda (\lambda +\lambda _{2})\eta _{2}\mu _{1L}\mu _{1Y}^{2}}{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\mu _{2L}\mu _{2Y}-\lambda ^{2}\mu _{1L}^{2}\mu _{1Y}^{2}}\frac{A}{\gamma a^{2} }=m_{1} \frac{A}{\gamma a^{2} }, \end{aligned}$$
(7.7)
$$\begin{aligned}&q_{2}^{*}=\frac{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\eta _{2}\mu _{1Y}\mu _{2L}-\lambda (\lambda +\lambda _{1})\eta _{1}\mu _{1L}^{2}\mu _{1Y}}{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\mu _{2L}\mu _{2Y}-\lambda ^{2}\mu _{1L}^{2}\mu _{1Y}^{2}}\frac{A}{\gamma a^{2} }=m_{2} \frac{A}{\gamma a^{2} }, \end{aligned}$$
(7.8)

and

$$\begin{aligned} f(q_{1}^{*},q_{2}^{*})= -\frac{A^{2}}{2\gamma a^{2} }\frac{a_{1}^{2}b_{2}^{2}\eta _{1}^{2}+a_{2}^{2}b_{1}^{2}\eta _{2}^{2}-2{\bar{\rho }} a_{1}a_{2} b_{1}b_{2}\eta _{1}\eta _{2}}{b_{1}^{2}b_{2}^{2}(1-{\bar{\rho }}^{2})} =-\frac{A^{2}}{2\gamma a^{2} }{\tilde{A}}_{2} \end{aligned}$$

with \({\tilde{A}}_{2}\) given by (3.9).

Then we look for the equilibrium value function. Inserting \(u^{*}\), \(q_{1}^{*}\) and \(q_{2}^{*}\) into (7.5) yields

$$\begin{aligned}&A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t}+ \frac{1}{2\gamma a^{2}}\frac{\left[ \mu (A-2\gamma \sigma _{1}\sigma _{12}ab)-(\gamma \sigma _{1}\sigma _{12}ac+rA)\right] ^{2}}{\sigma _{1}^{2} }+\frac{A^{2}}{2\gamma a^{2} }{\tilde{A}}_{2} \\&\quad +\left[ rx+c_{3}\right] A+\kappa (\delta -\mu )(2B\mu +C)+(\sigma _{12}^{2}+\sigma _{2}^{2})B-\frac{\gamma }{2}(\sigma _{12}^{2} +\sigma _{2}^{2})(2b\mu +c)^{2} =0. \end{aligned}$$

By separating variables, we obtain the following ordinary differential equations

$$\begin{aligned}&\left\{ \begin{aligned}&A_{t}+rA=0,\\&A(T)=1; \end{aligned}\right. \\&\left\{ \begin{aligned}&B_{t}-2\kappa B-2\gamma (\sigma _{12}^{2}+\sigma _{2}^{2}) b^{2}+\frac{1}{2\gamma a^{2}}\frac{(A-2\gamma \sigma _{1}\sigma _{12}ab)^{2}}{\sigma _{1}^{2} }=0,\\&B(T)=0; \end{aligned}\right. \\&\left\{ \begin{aligned}&C_{t}-\kappa C+2\kappa \delta B-2\gamma (\sigma _{12}^{2}+\sigma _{2}^{2}) bc-\frac{1}{\gamma a^{2}}\frac{(A-2\gamma \sigma _{1}\sigma _{12}ab)(\gamma \sigma _{1}\sigma _{12}ac+rA)}{\sigma _{1}^{2} }=0,\\&C(T)=0; \end{aligned}\right. \\&\left\{ \begin{aligned}&D_{t}+c_{3}A+(\sigma _{12}^{2}+\sigma _{2}^{2})B+\kappa \delta C-\frac{\gamma }{2}(\sigma _{12}^{2} +\sigma _{2}^{2})c^{2}+\frac{1}{2\gamma a^{2} }\frac{(\gamma \sigma _{1}\sigma _{12}ac+rA)^{2}}{\sigma _{1}^{2}}+\frac{A^{2}}{2\gamma a^{2} }{\tilde{A}}_{2}=0,\\&D(T)=0. \end{aligned}\right. \end{aligned}$$

Then we have \(A(t)=e^{r(T-t)}.\) Substituting the expression of \(u^{*}\), \(q_{1}^{*}\) and \(q_{2}^{*}\) given by (7.6), (7.7) and (7.8) into (7.4) yields

$$\begin{aligned}&a_{t}x\!+\!b_{t}\mu ^{2}\!+\!c_{t}\mu \!+\!d_{t}\!+ \! \left[ rx\!+\!\frac{(\mu -r)^{2}A\!-\!\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)(\mu -r)}{\gamma \sigma _{1}^{2}a^{2}}\!+\!c_{1}m_{1} \frac{A}{\gamma a^{2} }\!+\!c_{2}m_{2} \frac{A}{\gamma a^{2} }\!+\!c_{3}\!\right] \!a\\&\quad +\kappa (\delta -\mu )(2b\mu +c)+(\sigma _{12}^{2}+\sigma _{2}^{2})b-\frac{A}{\gamma a}(\lambda _{1}m_{1}\mu _{1L}+\lambda _{2}m_{2}\mu _{1Y})-\frac{A}{\gamma a}\lambda (m_{1}\mu _{1L}+m_{2}\mu _{1Y})=0. \end{aligned}$$

By separating variables again, we obtain

$$\begin{aligned}&a_{t}+ra=0,~a(T)=1;\\&b_{t}-2\kappa b+\frac{ A-2\gamma \sigma _{1}\sigma _{12}ab}{\gamma \sigma _{1}^{2}a}=0,~b(T)=0;\\&c_{t}-\left( \kappa +\frac{\sigma _{12}}{\sigma _{1}}\right) c+2\left( \kappa \delta +\frac{r\sigma _{12}}{\sigma _{1}}\right) b-2\frac{rA}{\gamma \sigma _{1}^{2}a}=0,~c(T)=0;\\&d_{t}\!+\!c_{3}a\!+\!(\!\sigma _{12}^{2}+\sigma _{2}^{2})b\!+\!\left( \!\kappa \delta \!+\!\frac{r\sigma _{12}}{\sigma _{1}}\right) \!c\!+\!\frac{A}{a}\!\bigg [\!\frac{r^{2}}{\gamma \sigma _{1}^{2}}\!+\!\frac{c_{1}m_{1}\!+\!c_{2}m_{2}\!-\!(\!\lambda \!+\!\lambda _{1}\!)m_{1}\mu _{1L}\!-\!(\!\lambda \!+\!\lambda _{2})m_{2}\mu _{1Y}}{\gamma } \!\bigg ]\!=\!0,\!\\&\quad d(T)=0. \end{aligned}$$

Then, we have \(a(t)=e^{r(T-t)},\) b(t) and c(t) are given by (3.13) and (3.14), respectively, and

$$\begin{aligned} d(t)&=\int _{t}^{T}\bigg [ c_{3}a(\tau )+(\sigma _{12}^{2}+\sigma _{2}^{2})b(\tau )+\left( \kappa \delta +\frac{r\sigma _{12}}{\sigma _{1}}\right) c(\tau )\\&\quad +\frac{r^{2}}{\gamma \sigma _{1}^{2}}+\frac{ m_{1}(\lambda +\lambda _{1})\eta _{1}\mu _{1L}+m_{2}(\lambda +\lambda _{2})\eta _{2}\mu _{1Y} }{\gamma } \bigg ]d\tau . \end{aligned}$$

Moreover, we can obtain the expressions of B(t), C(t) and \(D_{2}(t)\) as (3.16)–(3.18). Note that in Case 2, we use \(D_{2}(t)\) to denote D(t). Recall that in Case 2, \(m_{1}>0,~m_{2}>0\), so the equivalent reinsurance strategy \(q_{1}^{*},~q_{2}^{*}\) we have obtained in (7.7) and (7.8) satisfy the nonnegative constraint for the reinsurance strategy. The results in Case 1 and Case 3 can be derived similarly. This completes the proof. \(\square \)

Appendix B: The proof of Theorem 2

Proposition 2

The extended HJB system of Eq. (3.4) with \({\varPi }_{c}\), \(V_{c}\) and \(g_{c}\) replaced by \({\varPi }\), V and g, respectively, can be simplified as follows

$$\begin{aligned}&V_{t}+ \mathop {\sup }_{\pi \in {\varPi }}\bigg \{\left[ rx+({\widehat{\mu }}-r)u+c_{1}q_{1}+c_{2}q_{2}+c_{3}\right] V_{x}+\kappa (\delta -{\widehat{\mu }})V_{{\widehat{\mu }}} +\frac{1}{2}\sigma _{1}^{2} u^{2}V_{xx}\\&\quad +\frac{1}{2}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ^{2}V_{{\widehat{\mu }}{\widehat{\mu }}}\\&\quad +\sigma _{1}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) u V_{x{\widehat{\mu }}}-\frac{\gamma }{2}\sigma _{1}^{2} u^{2}g_{x}^{2}-\frac{\gamma }{2}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ^{2}g_{{\widehat{\mu }}}^{2}-\gamma \sigma _{1}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ug_{x}g_{{\widehat{\mu }}}\\&\quad +\lambda _{1}{\mathbb {E}}\left[ V(t,x-q_{1}L,{\widehat{\mu }})-V(t,x,{\widehat{\mu }})-\frac{\gamma }{2}(g(t,x-q_{1}L,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) )^{2} \right] \\&\quad +\lambda _{2}{\mathbb {E}}\left[ V(t,x-q_{2}Y,{\widehat{\mu }})-V(t,x,{\widehat{\mu }})-\frac{\gamma }{2}(g(t,x-q_{2}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) )^{2} \right] \\&\quad +\lambda {\mathbb {E}}\left[ V(t,x-q_{1}L-q_{2}Y,{\widehat{\mu }})-V(t,x,{\widehat{\mu }})-\frac{\gamma }{2}(g(t,x-q_{1}L-q_{2}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) )^{2} \right] \bigg \}=0, \\&V(T,x,{\widehat{\mu }})=x,\\&g_{t}+ \left[ rx+({\widehat{\mu }}-r){\widehat{u}}^{*}+c_{1}{\widehat{q}}_{1}^{*}+c_{2}{\widehat{q}}_{2}^{*}+c_{3}\right] g_{x}+\kappa (\delta -{\widehat{\mu }})g_{{\widehat{\mu }}} +\frac{1}{2}\sigma _{1}^{2} ({\widehat{u}}^{*})^{2}g_{xx}+\frac{1}{2}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ^{2}g_{{\widehat{\mu }}{\widehat{\mu }}} \\&\quad +\sigma _{1}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) {\widehat{u}}^{*} g_{x{\widehat{\mu }}}+\lambda _{1}{\mathbb {E}}\left[ g(t,x-{\widehat{q}}_{1}^{*}L,{\widehat{\mu }})-g(t,x,{\widehat{\mu }})\right] \\&\quad +\lambda _{2}{\mathbb {E}}\left[ g(t,x-{\widehat{q}}_{2}^{*}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) \right] \\&\quad +\lambda {\mathbb {E}}\left[ g(t,x-{\widehat{q}}_{1}^{*}L-{\widehat{q}}_{2}^{*}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) \right] =0, \\&g(T,x,{\widehat{\mu }})=x. \end{aligned}$$

Proof

The proof of this proposition is similar to that of Proposition 1 and thus is omitted. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bi, J., Cai, J. & Zeng, Y. Equilibrium reinsurance-investment strategies with partial information and common shock dependence. Ann Oper Res 307, 1–24 (2021). https://doi.org/10.1007/s10479-021-04317-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-021-04317-4

Keywords

Mathematics Subject Classification

Search

Navigation