American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2019003

Robust optimal investment and reinsurance of an insurer under Jump-diffusion models

 1 School of Mathematics, Southeast University, Nanjing, Jiangsu Province, 211189, China 2 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China 3 Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong Province, 518055, China 4 Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

* Corresponding author: Xin Zhang

Received  August 2017 Revised  April 2018 Published  August 2018

Fund Project: X. Zhang is supported by the National Natural Science Foundation of China (grant nos. 11771079, 11371020). H. Meng is supported by the National Natural Science Foundation of China (grant no. 11771465), the Program for Innovation Research in Central University of Finance and Economics, and the 111 Project (grant no. B17050). J. Xiong is supported by Southern University of Science and Technology startup fund (grant No. 28/Y01286120). Y. Shen is supported by the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2016-05677)

This paper studies a robust optimal investment and reinsurance problem under model uncertainty. The insurer's risk process is modeled by a general jump process generated by a marked point process. By transferring a proportion of insurance risk to a reinsurance company and investing the surplus into the financial market with a bond and a share index, the insurance company aims to maximize the minimal expected terminal wealth with a penalty. By using the dynamic programming, we formulate the robust optimal investment and reinsurance problem into a two-person, zero-sum, stochastic differential game between the investor and the market. Closed-form solutions for the case of the quadratic penalty function are derived in our paper.

Citation: Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019003
References:
 [1] S. Asmussen, B. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075. [2] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15. doi: 10.1016/S0167-6687(96)00017-0. [3] N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165. doi: 10.1007/s00186-005-0446-1. [4] N. Branger and L.S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, Journal of Banking and Finance, 37 (2013), 5036-5047. doi: 10.1016/j.jbankfin.2013.08.023. [5] 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. [6] S. Browne, Survival and growth with a liability: Optimal portfolio strategies in continuous time, Mathematics of Operations Research, 22 (1997), 468-493. doi: 10.1287/moor.22.2.468. [7] S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3 (1999), 275-294. doi: 10.1007/s007800050063. [8] A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330. doi: 10.1016/S0167-6687(00)00055-X. [9] Á. Cartea and S. Jaimungal, Risk metrics and fine tuning of high-frequency trading strategies, Mathematical Finance, 25 (2015), 576-611. doi: 10.1111/mafi.12023. [10] T. Choulli, M. Taksar and X. 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. [11] R. Cont, Model uncertanity and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2006), 519-547. doi: 10.1111/j.1467-9965.2006.00281.x. [12] J. Dupačová and J. Polívka, Asset-liability management for czech pension funds using stochastic programming, Annals of Operations Research, 165 (2009), 5-28. doi: 10.1007/s10479-008-0358-6. [13] R. Elliott, Stochastic Calculus and Applications, Springer Verlag, New York, 1982. [14] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. [15] 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. [16] C. Hipp and M. Taksar, Stochastic control for optimal new business, Insurance: Mathematics and Economics, 26 (2000), 185-192. doi: 10.1016/S0167-6687(99)00052-9. [17] B. Højgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182. doi: 10.1111/1467-9965.00066. [18] Z. Liang, Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Mathematica Sinica, Chinese Series, 51 (2008), 1195-1204. [19] X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the cev model, North American Actuarial Journal, 15 (2011), 417-431. doi: 10.1080/10920277.2011.10597628. [20] C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31. doi: 10.1080/10920277.2004.10596134. [21] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002. [22] F. Maccheroni, M. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498. doi: 10.1111/j.1468-0262.2006.00716.x. [23] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983. doi: 10.1093/rfs/hhh003. [24] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337. doi: 10.1080/17442500701655408. [25] H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies, Econocmic Modelling, 28 (2011), 211-218. doi: 10.1016/j.econmod.2010.09.009. [26] H. Meng, T. Siu and H. Yang, Optimal dividends with debts and nonlinear insurance risk processes, Insurance: Mathematics and Economics, 53 (2013), 110-121. doi: 10.1016/j.insmatheco.2013.04.008. [27] C. Moallemi and M. Sağlam, Dynamic portfolio choice with linear rebalancing rules, Journal of Financial and Quantitative Analysis, 52 (2017), 1247-1278. [28] National Association of Insurance Commissioners, Capital Markets Special Report: U. S. Insurance Industry Cash and Invested Assets at Year-End 2016. Available at http://www.naic.org/capital_markets_archive/170824.htm(2007). [29] B. Øksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Mathematical Finance, 19 (2009), 619-637. doi: 10.1111/j.1467-9965.2009.00382.x. [30] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68. doi: 10.1080/034612301750077338. [31] H. Schmidli, On minimising the ruin probability by investment and reinsurance, Annal of Applied Probability, 12 (2002), 890-907. doi: 10.1214/aoap/1031863173. [32] Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686. doi: 10.1016/j.jmaa.2016.09.053. [33] 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. [34] Z. Wen, X. Wu and Y. Zhou, Dynamic ordering policies under partial trade credit financing, in Service Systems and Service Management (ICSSSM), 2014 11th International Conference on, IEEE, 2014, 1-6. doi: 10.1109/ICSSSM.2014.6874077. [35] H. Yang and L. 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. [36] B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scandinavian Actuarial Journal, 2015 (2015), 725-751. doi: 10.1080/03461238.2014.883085. [37] C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773. doi: 10.1016/j.insmatheco.2013.09.019. [38] C. Yin, Y. Wen and Y. Zhao, On the optimal dividend problem for a spectrally positive levy process, ASTIN Bulletin, 44 (2014), 635-651. doi: 10.1017/asb.2014.12. [39] V.R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126. doi: 10.1080/10920277.2004.10596174. [40] J. Zhang and Q. Xiao, Optimal investment of a time-dependent renewal risk model with stochastic return, Journal of Inequalities and Applications, 2015 (2015), 12pp. doi: 10.1186/s13660-015-0707-3. [41] X. Zhang and T. 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. [42] X. Zhang, H. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean--variance premium principle and no-short selling, Insurance: Mathematics and Economics, 67 (2016), 125-132. doi: 10.1016/j.insmatheco.2016.01.001. [43] X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a cev model, Insurance: Mathematics and Economics, 67 (2016), 77-87. doi: 10.1016/j.insmatheco.2015.12.008. [44] M. Zhou and K. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207. doi: 10.1016/j.econmod.2011.09.007.

show all references

References:
 [1] S. Asmussen, B. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075. [2] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15. doi: 10.1016/S0167-6687(96)00017-0. [3] N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165. doi: 10.1007/s00186-005-0446-1. [4] N. Branger and L.S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, Journal of Banking and Finance, 37 (2013), 5036-5047. doi: 10.1016/j.jbankfin.2013.08.023. [5] 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. [6] S. Browne, Survival and growth with a liability: Optimal portfolio strategies in continuous time, Mathematics of Operations Research, 22 (1997), 468-493. doi: 10.1287/moor.22.2.468. [7] S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3 (1999), 275-294. doi: 10.1007/s007800050063. [8] A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330. doi: 10.1016/S0167-6687(00)00055-X. [9] Á. Cartea and S. Jaimungal, Risk metrics and fine tuning of high-frequency trading strategies, Mathematical Finance, 25 (2015), 576-611. doi: 10.1111/mafi.12023. [10] T. Choulli, M. Taksar and X. 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. [11] R. Cont, Model uncertanity and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2006), 519-547. doi: 10.1111/j.1467-9965.2006.00281.x. [12] J. Dupačová and J. Polívka, Asset-liability management for czech pension funds using stochastic programming, Annals of Operations Research, 165 (2009), 5-28. doi: 10.1007/s10479-008-0358-6. [13] R. Elliott, Stochastic Calculus and Applications, Springer Verlag, New York, 1982. [14] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. [15] 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. [16] C. Hipp and M. Taksar, Stochastic control for optimal new business, Insurance: Mathematics and Economics, 26 (2000), 185-192. doi: 10.1016/S0167-6687(99)00052-9. [17] B. Højgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182. doi: 10.1111/1467-9965.00066. [18] Z. Liang, Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Mathematica Sinica, Chinese Series, 51 (2008), 1195-1204. [19] X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the cev model, North American Actuarial Journal, 15 (2011), 417-431. doi: 10.1080/10920277.2011.10597628. [20] C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31. doi: 10.1080/10920277.2004.10596134. [21] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002. [22] F. Maccheroni, M. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498. doi: 10.1111/j.1468-0262.2006.00716.x. [23] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983. doi: 10.1093/rfs/hhh003. [24] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337. doi: 10.1080/17442500701655408. [25] H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies, Econocmic Modelling, 28 (2011), 211-218. doi: 10.1016/j.econmod.2010.09.009. [26] H. Meng, T. Siu and H. Yang, Optimal dividends with debts and nonlinear insurance risk processes, Insurance: Mathematics and Economics, 53 (2013), 110-121. doi: 10.1016/j.insmatheco.2013.04.008. [27] C. Moallemi and M. Sağlam, Dynamic portfolio choice with linear rebalancing rules, Journal of Financial and Quantitative Analysis, 52 (2017), 1247-1278. [28] National Association of Insurance Commissioners, Capital Markets Special Report: U. S. Insurance Industry Cash and Invested Assets at Year-End 2016. Available at http://www.naic.org/capital_markets_archive/170824.htm(2007). [29] B. Øksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Mathematical Finance, 19 (2009), 619-637. doi: 10.1111/j.1467-9965.2009.00382.x. [30] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68. doi: 10.1080/034612301750077338. [31] H. Schmidli, On minimising the ruin probability by investment and reinsurance, Annal of Applied Probability, 12 (2002), 890-907. doi: 10.1214/aoap/1031863173. [32] Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686. doi: 10.1016/j.jmaa.2016.09.053. [33] 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. [34] Z. Wen, X. Wu and Y. Zhou, Dynamic ordering policies under partial trade credit financing, in Service Systems and Service Management (ICSSSM), 2014 11th International Conference on, IEEE, 2014, 1-6. doi: 10.1109/ICSSSM.2014.6874077. [35] H. Yang and L. 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. [36] B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scandinavian Actuarial Journal, 2015 (2015), 725-751. doi: 10.1080/03461238.2014.883085. [37] C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773. doi: 10.1016/j.insmatheco.2013.09.019. [38] C. Yin, Y. Wen and Y. Zhao, On the optimal dividend problem for a spectrally positive levy process, ASTIN Bulletin, 44 (2014), 635-651. doi: 10.1017/asb.2014.12. [39] V.R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126. doi: 10.1080/10920277.2004.10596174. [40] J. Zhang and Q. Xiao, Optimal investment of a time-dependent renewal risk model with stochastic return, Journal of Inequalities and Applications, 2015 (2015), 12pp. doi: 10.1186/s13660-015-0707-3. [41] X. Zhang and T. 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. [42] X. Zhang, H. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean--variance premium principle and no-short selling, Insurance: Mathematics and Economics, 67 (2016), 125-132. doi: 10.1016/j.insmatheco.2016.01.001. [43] X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a cev model, Insurance: Mathematics and Economics, 67 (2016), 77-87. doi: 10.1016/j.insmatheco.2015.12.008. [44] M. Zhou and K. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207. doi: 10.1016/j.econmod.2011.09.007.
Effects of $\lambda$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
Effects of $q$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
Effects of $\beta_1$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
Effects of $\beta_2$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
 [1] Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247 [2] Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-26. doi: 10.3934/jimo.2018044 [3] Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100 [4] Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-31. doi: 10.3934/jimo.2018141 [5] M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365 [6] Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control & Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21 [7] Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092 [8] Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069 [9] Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2018298 [10] Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028 [11] Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447 [12] Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385 [13] T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028 [14] Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47 [15] Wenxue Huang, Yuanyi Pan, Lihong Zheng. Proportional association based roi model. Big Data & Information Analytics, 2017, 2 (2) : 119-125. doi: 10.3934/bdia.2017004 [16] Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 [17] Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2018, 8 (0) : 1-20. doi: 10.3934/mcrf.2019013 [18] Wei Wang, Linyi Qian, Xiaonan Su. Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model. Journal of Industrial & Management Optimization, 2015, 11 (2) : 493-514. doi: 10.3934/jimo.2015.11.493 [19] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [20] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

2017 Impact Factor: 0.631

Tools

Article outline

Figures and Tables