 Previous Article
 MCRF Home
 This Issue

Next Article
Relative controllability of linear systems of fractional order with delay
Stochastic recursive optimal control problem with time delay and applications
1.  School of Mathematics, Shandong University, Jinan 250100, China 
2.  School of Control Science and Engineering, Shandong University, Jinan 250061, China, China 
References:
[1] 
N. Agram, S. Haadem, B. Øksendal and F. Proske, A maximum principle for infinite horizon delay equations, SIAM Journal on Mathematical Analysis, 45 (2013), 24992522. doi: 10.1137/120882809. 
[2] 
N. Agram and B. Øksendal, Infinite horizon optimal control of forwardbackward stochastic differential equations with delay, Journal of Computational and Applied Mathematics, Series B, 259 (2014), 336349. doi: 10.1016/j.cam.2013.04.048. 
[3] 
M. Arriojas, Y. Z. Hu, S. E. A. Monhammed and G. Pap, A delayed Black and Scholes formula, Stochastic Analysis and Applications, 25 (2007), 471492. doi: 10.1080/07362990601139669. 
[4] 
K. Bahlali, F. Chighoub and B. Mezerdi, On the relationship between the stochastic maximum principle and dynamic programming in singular stochastic control, Stochastics: An International Journal of Probability and Stochastic Processes, 84 (2012), 233249. doi: 10.1080/17442508.2010.522238. 
[5] 
M. H. Chang, T. Pang and Y. P. Yang, A stochastic portfolio optimization model with bounded memory, Mathematics in Operations Research, 36 (2011), 604619. doi: 10.1287/moor.1110.0508. 
[6] 
L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica, 46 (2010), 10741080. doi: 10.1016/j.automatica.2010.03.005. 
[7] 
L. Chen and Z. Wu, Dynamic programming principle for stochastic recursive optimal control problem with delayed systems, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 10051026. doi: 10.1051/cocv/2011187. 
[8] 
F. Chighoub and B. Mezerdi, The relationship between the stochastic maximum principle and the dynamic programming in singular control of jump diffusions, International Journal of Stochastic Analysis, (2014), Article ID 201491, 17 pages. doi: 10.1155/2014/201491. 
[9] 
C. Donnelly, Suffcient stochastic maximum principle in a regimeswitching diffusion model, Applied Mathematics and Optimization, 64 (2011), 155169. doi: 10.1007/s0024501091309. 
[10] 
D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353394. doi: 10.2307/2951600. 
[11] 
N. El Karoui, S. G. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 171. doi: 10.1111/14679965.00022. 
[12] 
N. El Karoui, S. G. Peng and M. C. Quenez, A dynamic maximum principle for the optimization of recursive utilities under constraints, The Annals of Applied Probability, 11 (2001), 664693. doi: 10.1214/aoap/1015345345. 
[13] 
I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics & Stochastics Reports, 71 (2000), 6989. doi: 10.1080/17442500008834259. 
[14] 
S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance & Stochastics, 15 (2011), 421459. doi: 10.1007/s0078001001464. 
[15] 
N. C. Framstad, B. Øksendal and A. Sulem, A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance, Journal of Optimization Theory and Applications, 121 (2004), 7798. doi: 10.1023/B:JOTA.0000026132.62934.96. 
[16] 
M. Fuhrman, F. Masiero and G. Tessitore, Stochastic equations with delay: Optimal control via BSDEs and regular solutions of HamiltonJacobiBellman equations, SIAM Journal on Control and Optimization, 48 (2010), 46244651. doi: 10.1137/080730354. 
[17] 
F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, in Stochastic Partial Differential Equations and Applications VII (eds. G. Da Prato and L. Tubaro), Lecture Notes in Pure and Applied Mathematics, 245, Chapman & Hall, London, 2006, 133148. doi: 10.1201/9781420028720.ch13. 
[18] 
V. B. Kolmanovskii and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect, Automation & Remote Control, 34 (1973), 3952. 
[19] 
V. B. Kolmanovskii and L. E. Shaikhet, Control of Systems with Aftereffect, Translation of Mathematical Monographs, 157, American Mathematical Society, 1996. 
[20] 
B. Larssen, Dynamic programming in stochastic control of systems with delay, Stochastics & Stochastics Reports, 74 (2002), 651673. doi: 10.1080/1045112021000060764. 
[21] 
B. Larssen and N. H. Risebro, When are HJBequations in stochastic control of delay systems finite dimensional?, Stochastic Analysis and Applications, 21 (2003), 643671. doi: 10.1081/SAP120020430. 
[22] 
X. R. Mao and S. Sabanis, Delay geometric Brownian motion in financial option valuation, Stochastics: An International Journal of Probability and Stochastic Processes, 85 (2013), 295320. doi: 10.1080/17442508.2011.652965. 
[23] 
S. E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics VI, The Geido Workshop, 1996, Progress in Probability, 42, Birkhäuser, 1998, 177. 
[24] 
B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, in Optimal Control and Partial Differential Equations  Innovations and Applications (eds. J. M. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 2000. 
[25] 
B. Øksendal and A. Sulem, Maximum principles for optimal control of forwardbackward stochastic differential equations with jumps, SIAM Journal on Control and Optimization, 48 (2009), 29452976. doi: 10.1137/080739781. 
[26] 
B. Øksendal and A. Sulem, Forwardbackward stochastic differential games and stochastic control under model uncertainty, Journal of Optimization Theory and Applications, 161 (2014), 2255. doi: 10.1007/s1095701201667. 
[27] 
B. Øksendal, A. Sulem and T. S. Zhang, Optimal control of stochastic delay equations and timeadvanced backward stochastic differential equations, Advances in Applied Probability, 43 (2011), 572596. doi: 10.1239/aap/1308662493. 
[28] 
O. M. Pamen, Optimal control for stochastic delay system under model uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, (2013), published online. doi: 10.1007/s1095701304844. 
[29] 
E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 5561. doi: 10.1016/01676911(90)900826. 
[30] 
S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations, The Annals of Probability, 37 (2009), 877902. doi: 10.1214/08AOP423. 
[31] 
M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with stochastic differential utility, Journal of Economic Theory, 89 (1999), 68126. doi: 10.1006/jeth.1999.2558. 
[32] 
J. T. Shi, Relationship between maximum principle and dynamic programming for stochastic control systems with delay, in Proceedings of the 8th Asian Control Conference, Kaohsiung, Taiwan, 2011, 12101215. 
[33] 
J. T. Shi and Z. Wu, Maximum principle for forwardbackward stochastic control systems with random jumps and applications to finance, Journal of Systems Science and Complexity, 23 (2010), 219231. doi: 10.1007/s1142401072248. 
[34] 
J. T. Shi and Z. Wu, Relationship between MP and DPP for the optimal control problem of jump diffusions, Applied Mathematics and Optimization, 63 (2011), 151189. doi: 10.1007/s0024501091158. 
[35] 
J. T. Shi and Z. Y. Yu, Relationship between maximum principle and dynamic programming for stochastic recursive optimal control problems and applications, Mathematical Problems in Engineering, (2013), Article ID 285241, 12 pages. 
[36] 
G. C. Wang and Z. Wu, The maximum principle for stochastic recursive optimal control problems under partial information, IEEE Transactions on Automatic Control, 54 (2009), 12301242. doi: 10.1109/TAC.2009.2019794. 
[37] 
H. X. Wang and H. S. Zhang, LQ control for Itô type stochastic systems with input delays, Automatica, 49 (2013), 35383549. doi: 10.1016/j.automatica.2013.09.018. 
[38] 
J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, SpringerVerlag, New York, 1999. doi: 10.1007/9781461214663. 
[39] 
Z. Y. Yu, The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls, Automatica, 48 (2012), 24202432. doi: 10.1016/j.automatica.2012.06.082. 
[40] 
H. S. Zhang, G. R. Duan and L. H. Xie, Linear quadratic regulation for linear timevarying systems with multiple input delays, Automatica, 42 (2006), 14651476. doi: 10.1016/j.automatica.2006.04.007. 
[41] 
H. S. Zhang, G. Feng and C. Y. Han, Linear estimation for random delay systems, Systems & Control Letters, 60 (2011), 450459. doi: 10.1016/j.sysconle.2011.03.009. 
[42] 
X. Zhang, R. J. Elliott and T. K. Siu, A stochastic maximum principle for a Markov regimeswitching jumpdiffusion model and its application to finance, SIAM Journal on Control and Optimization, 50 (2012), 964990. doi: 10.1137/110839357. 
show all references
References:
[1] 
N. Agram, S. Haadem, B. Øksendal and F. Proske, A maximum principle for infinite horizon delay equations, SIAM Journal on Mathematical Analysis, 45 (2013), 24992522. doi: 10.1137/120882809. 
[2] 
N. Agram and B. Øksendal, Infinite horizon optimal control of forwardbackward stochastic differential equations with delay, Journal of Computational and Applied Mathematics, Series B, 259 (2014), 336349. doi: 10.1016/j.cam.2013.04.048. 
[3] 
M. Arriojas, Y. Z. Hu, S. E. A. Monhammed and G. Pap, A delayed Black and Scholes formula, Stochastic Analysis and Applications, 25 (2007), 471492. doi: 10.1080/07362990601139669. 
[4] 
K. Bahlali, F. Chighoub and B. Mezerdi, On the relationship between the stochastic maximum principle and dynamic programming in singular stochastic control, Stochastics: An International Journal of Probability and Stochastic Processes, 84 (2012), 233249. doi: 10.1080/17442508.2010.522238. 
[5] 
M. H. Chang, T. Pang and Y. P. Yang, A stochastic portfolio optimization model with bounded memory, Mathematics in Operations Research, 36 (2011), 604619. doi: 10.1287/moor.1110.0508. 
[6] 
L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica, 46 (2010), 10741080. doi: 10.1016/j.automatica.2010.03.005. 
[7] 
L. Chen and Z. Wu, Dynamic programming principle for stochastic recursive optimal control problem with delayed systems, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 10051026. doi: 10.1051/cocv/2011187. 
[8] 
F. Chighoub and B. Mezerdi, The relationship between the stochastic maximum principle and the dynamic programming in singular control of jump diffusions, International Journal of Stochastic Analysis, (2014), Article ID 201491, 17 pages. doi: 10.1155/2014/201491. 
[9] 
C. Donnelly, Suffcient stochastic maximum principle in a regimeswitching diffusion model, Applied Mathematics and Optimization, 64 (2011), 155169. doi: 10.1007/s0024501091309. 
[10] 
D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353394. doi: 10.2307/2951600. 
[11] 
N. El Karoui, S. G. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 171. doi: 10.1111/14679965.00022. 
[12] 
N. El Karoui, S. G. Peng and M. C. Quenez, A dynamic maximum principle for the optimization of recursive utilities under constraints, The Annals of Applied Probability, 11 (2001), 664693. doi: 10.1214/aoap/1015345345. 
[13] 
I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics & Stochastics Reports, 71 (2000), 6989. doi: 10.1080/17442500008834259. 
[14] 
S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance & Stochastics, 15 (2011), 421459. doi: 10.1007/s0078001001464. 
[15] 
N. C. Framstad, B. Øksendal and A. Sulem, A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance, Journal of Optimization Theory and Applications, 121 (2004), 7798. doi: 10.1023/B:JOTA.0000026132.62934.96. 
[16] 
M. Fuhrman, F. Masiero and G. Tessitore, Stochastic equations with delay: Optimal control via BSDEs and regular solutions of HamiltonJacobiBellman equations, SIAM Journal on Control and Optimization, 48 (2010), 46244651. doi: 10.1137/080730354. 
[17] 
F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, in Stochastic Partial Differential Equations and Applications VII (eds. G. Da Prato and L. Tubaro), Lecture Notes in Pure and Applied Mathematics, 245, Chapman & Hall, London, 2006, 133148. doi: 10.1201/9781420028720.ch13. 
[18] 
V. B. Kolmanovskii and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect, Automation & Remote Control, 34 (1973), 3952. 
[19] 
V. B. Kolmanovskii and L. E. Shaikhet, Control of Systems with Aftereffect, Translation of Mathematical Monographs, 157, American Mathematical Society, 1996. 
[20] 
B. Larssen, Dynamic programming in stochastic control of systems with delay, Stochastics & Stochastics Reports, 74 (2002), 651673. doi: 10.1080/1045112021000060764. 
[21] 
B. Larssen and N. H. Risebro, When are HJBequations in stochastic control of delay systems finite dimensional?, Stochastic Analysis and Applications, 21 (2003), 643671. doi: 10.1081/SAP120020430. 
[22] 
X. R. Mao and S. Sabanis, Delay geometric Brownian motion in financial option valuation, Stochastics: An International Journal of Probability and Stochastic Processes, 85 (2013), 295320. doi: 10.1080/17442508.2011.652965. 
[23] 
S. E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics VI, The Geido Workshop, 1996, Progress in Probability, 42, Birkhäuser, 1998, 177. 
[24] 
B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, in Optimal Control and Partial Differential Equations  Innovations and Applications (eds. J. M. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 2000. 
[25] 
B. Øksendal and A. Sulem, Maximum principles for optimal control of forwardbackward stochastic differential equations with jumps, SIAM Journal on Control and Optimization, 48 (2009), 29452976. doi: 10.1137/080739781. 
[26] 
B. Øksendal and A. Sulem, Forwardbackward stochastic differential games and stochastic control under model uncertainty, Journal of Optimization Theory and Applications, 161 (2014), 2255. doi: 10.1007/s1095701201667. 
[27] 
B. Øksendal, A. Sulem and T. S. Zhang, Optimal control of stochastic delay equations and timeadvanced backward stochastic differential equations, Advances in Applied Probability, 43 (2011), 572596. doi: 10.1239/aap/1308662493. 
[28] 
O. M. Pamen, Optimal control for stochastic delay system under model uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, (2013), published online. doi: 10.1007/s1095701304844. 
[29] 
E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 5561. doi: 10.1016/01676911(90)900826. 
[30] 
S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations, The Annals of Probability, 37 (2009), 877902. doi: 10.1214/08AOP423. 
[31] 
M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with stochastic differential utility, Journal of Economic Theory, 89 (1999), 68126. doi: 10.1006/jeth.1999.2558. 
[32] 
J. T. Shi, Relationship between maximum principle and dynamic programming for stochastic control systems with delay, in Proceedings of the 8th Asian Control Conference, Kaohsiung, Taiwan, 2011, 12101215. 
[33] 
J. T. Shi and Z. Wu, Maximum principle for forwardbackward stochastic control systems with random jumps and applications to finance, Journal of Systems Science and Complexity, 23 (2010), 219231. doi: 10.1007/s1142401072248. 
[34] 
J. T. Shi and Z. Wu, Relationship between MP and DPP for the optimal control problem of jump diffusions, Applied Mathematics and Optimization, 63 (2011), 151189. doi: 10.1007/s0024501091158. 
[35] 
J. T. Shi and Z. Y. Yu, Relationship between maximum principle and dynamic programming for stochastic recursive optimal control problems and applications, Mathematical Problems in Engineering, (2013), Article ID 285241, 12 pages. 
[36] 
G. C. Wang and Z. Wu, The maximum principle for stochastic recursive optimal control problems under partial information, IEEE Transactions on Automatic Control, 54 (2009), 12301242. doi: 10.1109/TAC.2009.2019794. 
[37] 
H. X. Wang and H. S. Zhang, LQ control for Itô type stochastic systems with input delays, Automatica, 49 (2013), 35383549. doi: 10.1016/j.automatica.2013.09.018. 
[38] 
J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, SpringerVerlag, New York, 1999. doi: 10.1007/9781461214663. 
[39] 
Z. Y. Yu, The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls, Automatica, 48 (2012), 24202432. doi: 10.1016/j.automatica.2012.06.082. 
[40] 
H. S. Zhang, G. R. Duan and L. H. Xie, Linear quadratic regulation for linear timevarying systems with multiple input delays, Automatica, 42 (2006), 14651476. doi: 10.1016/j.automatica.2006.04.007. 
[41] 
H. S. Zhang, G. Feng and C. Y. Han, Linear estimation for random delay systems, Systems & Control Letters, 60 (2011), 450459. doi: 10.1016/j.sysconle.2011.03.009. 
[42] 
X. Zhang, R. J. Elliott and T. K. Siu, A stochastic maximum principle for a Markov regimeswitching jumpdiffusion model and its application to finance, SIAM Journal on Control and Optimization, 50 (2012), 964990. doi: 10.1137/110839357. 
[1] 
Haiyang Wang, Zhen Wu. Timeinconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control and Related Fields, 2015, 5 (3) : 651678. doi: 10.3934/mcrf.2015.5.651 
[2] 
Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1. doi: 10.1186/s4154601700147 
[3] 
Ishak Alia. Timeinconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control and Related Fields, 2020, 10 (4) : 785826. doi: 10.3934/mcrf.2020020 
[4] 
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437446. doi: 10.3934/proc.2013.2013.437 
[5] 
Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations and Control Theory, 2014, 3 (1) : 3558. doi: 10.3934/eect.2014.3.35 
[6] 
Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$scheme for solving backward stochastic differential equations. Discrete and Continuous Dynamical Systems  B, 2012, 17 (5) : 15851603. doi: 10.3934/dcdsb.2012.17.1585 
[7] 
Zhen Wu, Feng Zhang. Maximum principle for discretetime stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475493. doi: 10.3934/mcrf.2021031 
[8] 
Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 52215237. doi: 10.3934/dcds.2015.35.5221 
[9] 
Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risksensitive of fully coupled forwardbackward stochastic control of meanfield type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817843. doi: 10.3934/eect.2020035 
[10] 
Editorial Office. Retraction: XiaoQian Jiang and LunChuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete and Continuous Dynamical Systems  S, 2019, 12 (4&5) : 969969. doi: 10.3934/dcdss.2019065 
[11] 
Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 615635. doi: 10.3934/dcdsb.2018199 
[12] 
Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems  B, 2016, 21 (9) : 30153027. doi: 10.3934/dcdsb.2016085 
[13] 
András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks and Heterogeneous Media, 2012, 7 (1) : 4358. doi: 10.3934/nhm.2012.7.43 
[14] 
Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete and Continuous Dynamical Systems  B, 2013, 18 (6) : 16831696. doi: 10.3934/dcdsb.2013.18.1683 
[15] 
Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with nonlipschitz coefficients. Discrete and Continuous Dynamical Systems  B, 2019, 24 (7) : 32993318. doi: 10.3934/dcdsb.2018321 
[16] 
Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$Brownian motion. Discrete and Continuous Dynamical Systems  B, 2015, 20 (1) : 281293. doi: 10.3934/dcdsb.2015.20.281 
[17] 
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blowup problems for a stochastic differential equation. Discrete and Continuous Dynamical Systems  S, 2021, 14 (3) : 909918. doi: 10.3934/dcdss.2020391 
[18] 
Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete and Continuous Dynamical Systems  B, 2021, 26 (9) : 48874905. doi: 10.3934/dcdsb.2020317 
[19] 
Hancheng Guo, Jie Xiong. A secondorder stochastic maximum principle for generalized meanfield singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451473. doi: 10.3934/mcrf.2018018 
[20] 
Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 22892331. doi: 10.3934/cpaa.2020100 
2020 Impact Factor: 1.284
Tools
Metrics
Other articles
by authors
[Back to Top]