Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps

This paper studies an optimal investment and reinsurance problem incorporating jumps for mean–variance insurers within a game theoretic framework and aims to seek the corresponding time-consistent strategies. Specially, the insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where the surplus of the insurers is assumed to follow a jump–diffusion model and the financial market consists of one risk-free asset and one risky asset whose price process is modeled by a geometric Levy process. By solving an extended Hamilton–Jacobi–Bellman system, the closed-form expressions for the time-consistent investment and reinsurance strategies and the optimal value function are derived. Moreover, some special cases of our model and results are presented, and some numerical illustrations and sensitivity analysis for our results are provided.

[1]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[2]  Esben Masotti Kryger,et al.  Some Solvable Portfolio Problems with Quadratic and Collective Objectives , 2010 .

[3]  J. Grandell.  Aspects of Risk Theory , 1991 .

[4]  Duan Li,et al.  Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation , 2000 .

[5]  Süleyman Özekici,et al.  Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach , 2007, Eur. J. Oper. Res..

[6]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[7]  Yan Zeng,et al.  Optimal reinsurance-investment strategies for insurers under mean-CaR criteria , 2012 .

[8]  Jorge Navas,et al.  Consumption and portfolio rules for time-inconsistent investors , 2010, Eur. J. Oper. Res..

[9]  Alex W. H. Chan.  Merton, Robert C. , 2010 .

[10]  X. Zhou,et al.  MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION , 2014 .

[11]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[12]  Xun Li,et al.  Dynamic mean-variance portfolio selection with borrowing constraint , 2010, Eur. J. Oper. Res..

[13]  Ivar Ekeland,et al.  The golden rule when preferences are time inconsistent , 2010 .

[14]  Søren Asmussen,et al.  Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation , 2000, Finance Stochastics.

[15]  Traian A. Pirvu,et al.  Time-Consistent Portfolio Management , 2010, SIAM J. Financial Math..

[16]  Huiling Wu,et al.  Multi-period mean–variance portfolio selection with regime switching and a stochastic cash flow☆ , 2012 .

[17]  Süleyman Özekici,et al.  Portfolio optimization in stochastic markets , 2006, Math. Methods Oper. Res..

[18]  Jorge Navas,et al.  Consumption, Investment and Life Insurance Strategies with Heterogeneous Discounting , 2012 .

[19]  J. Wang,et al.  Continuous time mean variance asset allocation: A time-consistent strategy , 2011, Eur. J. Oper. Res..

[20]  Ivar Ekeland,et al.  Equilibrium policies when preferences are time inconsistent , 2008, 0808.3790.

[21]  Nicole Bäuerle.  Benchmark and mean-variance problems for insurers , 2005, Math. Methods Oper. Res..

[22]  Zhongfei Li,et al.  Multi-period mean-variance portfolio selection with Markov regime switching and uncertain time-horizon , 2011, J. Syst. Sci. Complex..

[23]  Yan Zeng,et al.  Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model , 2012 .

[24]  X. Zhou,et al.  Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework , 2000 .

[25]  Heterogeneous discounting in economic problems , 2012 .

[26]  Jianming Xia,et al.  MARKOWITZ'S PORTFOLIO OPTIMIZATION IN AN INCOMPLETE MARKET , 2006 .

[27]  Suleyman Basak,et al.  Dynamic Mean-Variance Asset Allocation , 2009 .

[28]  Zhongfei Li,et al.  Optimal time-consistent investment and reinsurance policies for mean-variance insurers , 2011 .

[29]  I. Ekeland,et al.  Investment and consumption without commitment , 2007, 0708.0588.

[30]  Lukasz Delong,et al.  Mean-variance portfolio selection for a non-life insurance company , 2007, Math. Methods Oper. Res..

[31]  R. H. Strotz.  Myopia and Inconsistency in Dynamic Utility Maximization , 1955 .

[32]  Zhongfei Li,et al.  Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers , 2010 .

[33]  X. Zhou,et al.  CONTINUOUS‐TIME MEAN‐VARIANCE PORTFOLIO SELECTION WITH BANKRUPTCY PROHIBITION , 2005 .

[34]  Tomas Bjork,et al.  A General Theory of Markovian Time Inconsistent Stochastic Control Problems , 2010 .

[35]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[36]  Lihua Bai,et al.  Dynamic mean-variance problem with constrained risk control for the insurers , 2008, Math. Methods Oper. Res..