Optimal stochastic differential games with VaR constraints

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September 2013, 18(7): 1889-1907. doi: 10.3934/dcdsb.2013.18.1889

Optimal stochastic differential games with VaR constraints

Jingzhen Liu ^1, and Ka-Fai Cedric Yiu ^2,

1.	School of Insurance, Central University Of Finance and Economics, Beijing 100081, China
2.	Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received May 2011 Revised January 2013 Published May 2013

Abstract
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The nonlinear dynamic games between competing insurance companies are interesting and important problems because of the general practice of using re-insurance to reduce risks in the insurance industry. This problem becomes more complicated if a proper risk control is imposed on all the involving companies. In order to understand the dynamical properties, we consider the stochastic differential game between two insurance companies with risk constraints. The companies are allowed to purchase proportional reinsurance and invest their money into both risk free asset and risky (stock) asset. The competition between the two companies is formulated as a two player (zero-sum) stochastic differential game. One company chooses the optimal reinsurance and investment strategy in order to maximize the expected payoff, and the other one tries to minimize this value. For the purpose of risk management, the risk arising from the whole portfolio is constrained to some level. By the principle of dynamic programming, the problem is reduced to solving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations for Nash equilibria. We derive the Nash equilibria explicitly and obtain closed form solutions to HJBI under different scenarios.

Keywords: dynamic programming., risk constraint, stochastic differential game, HJBI equations, Nash equilibria, proportional reinsurance, Optimal investment.

Mathematics Subject Classification: Primary: 93E20, 91A25; Secondary: 49l2.

Citation: Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889

References:

[1]	N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems,", North Holland, (1981).
[2]	L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint,, Insurance Math. Econom., 42 (2008), 968. doi: 10.1016/j.insmatheco.2007.11.002.
[3]	N. Bauerle, Benchmark and mean-variance problems for insurers,, Mathematical Methods of Operations Research, 62 (2005), 159. doi: 10.1007/s00186-005-0446-1.
[4]	R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk,, in, (2000), 31.
[5]	S. Bouma, "Risk Management in the Insurance Industry and Solvency II,", European Survey, (2006).
[6]	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. doi: 10.1287/moor.20.4.937.
[7]	S. Browne, Stochastic differential portfolio games,, J. Appl. Prob., 37 (2000), 126. doi: 10.1239/jap/1014842273.
[8]	R. J. Elliott and T. K. Siu, On risk minimizing portfolio choice under Markovian regime-switching model,, Ann. Oper. Res., 176 (2010), 271. doi: 10.1007/s10479-008-0448-5.
[9]	R. J. Elliott and T. K. Siu, A stochastic differential game for optimal investment of an insurer with regime switching,, Quant. Finance, 11 (2011), 365. doi: 10.1080/14697681003591704.
[10]	B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models,, Scand. Actuar. J., 22 (1998), 166.
[11]	J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint,, Journal of Industrial and Management Optimization, 8 (2012), 531. doi: 10.3934/jimo.2012.8.531.
[12]	J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal Portfolios with stress analysis and the effect of a CVaR constraint,, Pac. J. Optim., 7 (2011), 83.
[13]	S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance Math. Econom., 42 (2008), 434. doi: 10.1016/j.insmatheco.2007.04.002.
[14]	B. Øksendal and A. Sulem, "A Game Theoretic Approach to Amrtingale Measures in Incomplete Markets,", University of Oslo and INRIA, (2006).
[15]	D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift,, North American Actuarial Journal, 9 (2005), 109.
[16]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting,, Scand. Actuar. J., 2001 (2001), 55. doi: 10.1080/034612301750077338.
[17]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, Ann. Appl. Probab., 12 (2002), 890. doi: 10.1214/aoap/1031863173.
[18]	J. Suijs, A. De Waegenaere and P. Borm, Stochastic cooperative games in insurance,, Insurance Math. Econom., 22 (1998), 209. doi: 10.1016/S0167-6687(97)00038-3.
[19]	K. L. Teo, D. W. Reid and I. E. Boyd, Stochastic optimal control theory and its computational method,, Internat. J. Systems Sci., 11 (1980), 77. doi: 10.1080/00207728008966998.
[20]	H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance Math. Econom., 37 (2005), 615. doi: 10.1016/j.insmatheco.2005.06.009.
[21]	K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint,, J. Econom. Dynam. Control, 28 (2004), 1317. doi: 10.1016/S0165-1889(03)00116-7.
[22]	K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint,, Automatica, 46 (2010), 979. doi: 10.1016/j.automatica.2010.02.027.
[23]	X. D. Zeng, A stochastic differential reinsurance game,, J. Appl. Prob., 47 (2010), 335. doi: 10.1239/jap/1276784895.
[24]	X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty,, Insurance Math. Econom., 45 (2009), 81. doi: 10.1016/j.insmatheco.2009.04.001.

show all references

References:

[1]	N. U. Ahmed and K. L. Teo, "Optimal Control of Distributed Parameter Systems,", North Holland, (1981).
[2]	L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint,, Insurance Math. Econom., 42 (2008), 968. doi: 10.1016/j.insmatheco.2007.11.002.
[3]	N. Bauerle, Benchmark and mean-variance problems for insurers,, Mathematical Methods of Operations Research, 62 (2005), 159. doi: 10.1007/s00186-005-0446-1.
[4]	R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk,, in, (2000), 31.
[5]	S. Bouma, "Risk Management in the Insurance Industry and Solvency II,", European Survey, (2006).
[6]	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. doi: 10.1287/moor.20.4.937.
[7]	S. Browne, Stochastic differential portfolio games,, J. Appl. Prob., 37 (2000), 126. doi: 10.1239/jap/1014842273.
[8]	R. J. Elliott and T. K. Siu, On risk minimizing portfolio choice under Markovian regime-switching model,, Ann. Oper. Res., 176 (2010), 271. doi: 10.1007/s10479-008-0448-5.
[9]	R. J. Elliott and T. K. Siu, A stochastic differential game for optimal investment of an insurer with regime switching,, Quant. Finance, 11 (2011), 365. doi: 10.1080/14697681003591704.
[10]	B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models,, Scand. Actuar. J., 22 (1998), 166.
[11]	J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint,, Journal of Industrial and Management Optimization, 8 (2012), 531. doi: 10.3934/jimo.2012.8.531.
[12]	J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal Portfolios with stress analysis and the effect of a CVaR constraint,, Pac. J. Optim., 7 (2011), 83.
[13]	S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance Math. Econom., 42 (2008), 434. doi: 10.1016/j.insmatheco.2007.04.002.
[14]	B. Øksendal and A. Sulem, "A Game Theoretic Approach to Amrtingale Measures in Incomplete Markets,", University of Oslo and INRIA, (2006).
[15]	D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift,, North American Actuarial Journal, 9 (2005), 109.
[16]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting,, Scand. Actuar. J., 2001 (2001), 55. doi: 10.1080/034612301750077338.
[17]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, Ann. Appl. Probab., 12 (2002), 890. doi: 10.1214/aoap/1031863173.
[18]	J. Suijs, A. De Waegenaere and P. Borm, Stochastic cooperative games in insurance,, Insurance Math. Econom., 22 (1998), 209. doi: 10.1016/S0167-6687(97)00038-3.
[19]	K. L. Teo, D. W. Reid and I. E. Boyd, Stochastic optimal control theory and its computational method,, Internat. J. Systems Sci., 11 (1980), 77. doi: 10.1080/00207728008966998.
[20]	H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance Math. Econom., 37 (2005), 615. doi: 10.1016/j.insmatheco.2005.06.009.
[21]	K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint,, J. Econom. Dynam. Control, 28 (2004), 1317. doi: 10.1016/S0165-1889(03)00116-7.
[22]	K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint,, Automatica, 46 (2010), 979. doi: 10.1016/j.automatica.2010.02.027.
[23]	X. D. Zeng, A stochastic differential reinsurance game,, J. Appl. Prob., 47 (2010), 335. doi: 10.1239/jap/1276784895.
[24]	X. Zhang and T. K. Siu, Optimal investment and reinsurance of an insurer with model uncertainty,, Insurance Math. Econom., 45 (2009), 81. doi: 10.1016/j.insmatheco.2009.04.001.

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