December  2015, 5(4): 859-888. doi: 10.3934/mcrf.2015.5.859

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

Received  August 2014 Revised  March 2015 Published  October 2015

This paper is concerned with a stochastic recursive optimal control problem with time delay, where the controlled system is described by a stochastic differential delayed equation (SDDE) and the cost functional is formulated as the solution to a backward SDDE (BSDDE). When there are only the pointwise and distributed time delays in the state variable, a generalized Hamilton-Jacobi-Bellman (HJB) equation for the value function in finite dimensional space is obtained, applying dynamic programming principle. This generalized HJB equation admits a smooth solution when the coefficients satisfy a particular system of first order partial differential equations (PDEs). A sufficient maximum principle is derived, where the adjoint equation is a forward-backward SDDE (FBSDDE). Under some differentiability assumptions, the relationship between the value function, the adjoint processes and the generalized Hamiltonian function is obtained. A consumption and portfolio optimization problem with recursive utility in the financial market, is discussed to show the applications of our result. Explicit solutions in a finite dimensional space derived by the two different approaches, coincide.
Citation: Jingtao Shi, Juanjuan Xu, Huanshui Zhang. Stochastic recursive optimal control problem with time delay and applications. Mathematical Control & Related Fields, 2015, 5 (4) : 859-888. doi: 10.3934/mcrf.2015.5.859
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), 2499. doi: 10.1137/120882809. Google Scholar

[2]

N. Agram and B. Øksendal, Infinite horizon optimal control of forward-backward stochastic differential equations with delay,, Journal of Computational and Applied Mathematics, 259 (2014), 336. doi: 10.1016/j.cam.2013.04.048. Google Scholar

[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), 471. doi: 10.1080/07362990601139669. Google Scholar

[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), 233. doi: 10.1080/17442508.2010.522238. Google Scholar

[5]

M. H. Chang, T. Pang and Y. P. Yang, A stochastic portfolio optimization model with bounded memory,, Mathematics in Operations Research, 36 (2011), 604. doi: 10.1287/moor.1110.0508. Google Scholar

[6]

L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application,, Automatica, 46 (2010), 1074. doi: 10.1016/j.automatica.2010.03.005. Google Scholar

[7]

L. Chen and Z. Wu, Dynamic programming principle for stochastic recursive optimal control problem with delayed systems,, ESAIM: Control, 18 (2012), 1005. doi: 10.1051/cocv/2011187. Google Scholar

[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). doi: 10.1155/2014/201491. Google Scholar

[9]

C. Donnelly, Suffcient stochastic maximum principle in a regime-switching diffusion model,, Applied Mathematics and Optimization, 64 (2011), 155. doi: 10.1007/s00245-010-9130-9. Google Scholar

[10]

D. Duffie and L. G. Epstein, Stochastic differential utility,, Econometrica, 60 (1992), 353. doi: 10.2307/2951600. Google Scholar

[11]

N. El Karoui, S. G. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1. doi: 10.1111/1467-9965.00022. Google Scholar

[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), 664. doi: 10.1214/aoap/1015345345. Google Scholar

[13]

I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay,, Stochastics & Stochastics Reports, 71 (2000), 69. doi: 10.1080/17442500008834259. Google Scholar

[14]

S. Federico, A stochastic control problem with delay arising in a pension fund model,, Finance & Stochastics, 15 (2011), 421. doi: 10.1007/s00780-010-0146-4. Google Scholar

[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), 77. doi: 10.1023/B:JOTA.0000026132.62934.96. Google Scholar

[16]

M. Fuhrman, F. Masiero and G. Tessitore, Stochastic equations with delay: Optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations,, SIAM Journal on Control and Optimization, 48 (2010), 4624. doi: 10.1137/080730354. Google Scholar

[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), (2006), 133. doi: 10.1201/9781420028720.ch13. Google Scholar

[18]

V. B. Kolmanovskii and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect,, Automation & Remote Control, 34 (1973), 39. Google Scholar

[19]

V. B. Kolmanovskii and L. E. Shaikhet, Control of Systems with Aftereffect,, Translation of Mathematical Monographs, (1996). Google Scholar

[20]

B. Larssen, Dynamic programming in stochastic control of systems with delay,, Stochastics & Stochastics Reports, 74 (2002), 651. doi: 10.1080/1045112021000060764. Google Scholar

[21]

B. Larssen and N. H. Risebro, When are HJB-equations in stochastic control of delay systems finite dimensional?,, Stochastic Analysis and Applications, 21 (2003), 643. doi: 10.1081/SAP-120020430. Google Scholar

[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), 295. doi: 10.1080/17442508.2011.652965. Google Scholar

[23]

S. E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications,, in Stochastic Analysis and Related Topics VI, (1996), 1. Google Scholar

[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, (2000). Google Scholar

[25]

B. Øksendal and A. Sulem, Maximum principles for optimal control of forward-backward stochastic differential equations with jumps,, SIAM Journal on Control and Optimization, 48 (2009), 2945. doi: 10.1137/080739781. Google Scholar

[26]

B. Øksendal and A. Sulem, Forward-backward stochastic differential games and stochastic control under model uncertainty,, Journal of Optimization Theory and Applications, 161 (2014), 22. doi: 10.1007/s10957-012-0166-7. Google Scholar

[27]

B. Øksendal, A. Sulem and T. S. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations,, Advances in Applied Probability, 43 (2011), 572. doi: 10.1239/aap/1308662493. Google Scholar

[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). doi: 10.1007/s10957-013-0484-4. Google Scholar

[29]

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation,, Systems & Control Letters, 14 (1990), 55. doi: 10.1016/0167-6911(90)90082-6. Google Scholar

[30]

S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations,, The Annals of Probability, 37 (2009), 877. doi: 10.1214/08-AOP423. Google Scholar

[31]

M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with stochastic differential utility,, Journal of Economic Theory, 89 (1999), 68. doi: 10.1006/jeth.1999.2558. Google Scholar

[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, (2011), 1210. Google Scholar

[33]

J. T. Shi and Z. Wu, Maximum principle for forward-backward stochastic control systems with random jumps and applications to finance,, Journal of Systems Science and Complexity, 23 (2010), 219. doi: 10.1007/s11424-010-7224-8. Google Scholar

[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), 151. doi: 10.1007/s00245-010-9115-8. Google Scholar

[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). Google Scholar

[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), 1230. doi: 10.1109/TAC.2009.2019794. Google Scholar

[37]

H. X. Wang and H. S. Zhang, LQ control for Itô type stochastic systems with input delays,, Automatica, 49 (2013), 3538. doi: 10.1016/j.automatica.2013.09.018. Google Scholar

[38]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

[39]

Z. Y. Yu, The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls,, Automatica, 48 (2012), 2420. doi: 10.1016/j.automatica.2012.06.082. Google Scholar

[40]

H. S. Zhang, G. R. Duan and L. H. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays,, Automatica, 42 (2006), 1465. doi: 10.1016/j.automatica.2006.04.007. Google Scholar

[41]

H. S. Zhang, G. Feng and C. Y. Han, Linear estimation for random delay systems,, Systems & Control Letters, 60 (2011), 450. doi: 10.1016/j.sysconle.2011.03.009. Google Scholar

[42]

X. Zhang, R. J. Elliott and T. K. Siu, A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance,, SIAM Journal on Control and Optimization, 50 (2012), 964. doi: 10.1137/110839357. Google Scholar

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), 2499. doi: 10.1137/120882809. Google Scholar

[2]

N. Agram and B. Øksendal, Infinite horizon optimal control of forward-backward stochastic differential equations with delay,, Journal of Computational and Applied Mathematics, 259 (2014), 336. doi: 10.1016/j.cam.2013.04.048. Google Scholar

[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), 471. doi: 10.1080/07362990601139669. Google Scholar

[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), 233. doi: 10.1080/17442508.2010.522238. Google Scholar

[5]

M. H. Chang, T. Pang and Y. P. Yang, A stochastic portfolio optimization model with bounded memory,, Mathematics in Operations Research, 36 (2011), 604. doi: 10.1287/moor.1110.0508. Google Scholar

[6]

L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application,, Automatica, 46 (2010), 1074. doi: 10.1016/j.automatica.2010.03.005. Google Scholar

[7]

L. Chen and Z. Wu, Dynamic programming principle for stochastic recursive optimal control problem with delayed systems,, ESAIM: Control, 18 (2012), 1005. doi: 10.1051/cocv/2011187. Google Scholar

[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). doi: 10.1155/2014/201491. Google Scholar

[9]

C. Donnelly, Suffcient stochastic maximum principle in a regime-switching diffusion model,, Applied Mathematics and Optimization, 64 (2011), 155. doi: 10.1007/s00245-010-9130-9. Google Scholar

[10]

D. Duffie and L. G. Epstein, Stochastic differential utility,, Econometrica, 60 (1992), 353. doi: 10.2307/2951600. Google Scholar

[11]

N. El Karoui, S. G. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1. doi: 10.1111/1467-9965.00022. Google Scholar

[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), 664. doi: 10.1214/aoap/1015345345. Google Scholar

[13]

I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay,, Stochastics & Stochastics Reports, 71 (2000), 69. doi: 10.1080/17442500008834259. Google Scholar

[14]

S. Federico, A stochastic control problem with delay arising in a pension fund model,, Finance & Stochastics, 15 (2011), 421. doi: 10.1007/s00780-010-0146-4. Google Scholar

[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), 77. doi: 10.1023/B:JOTA.0000026132.62934.96. Google Scholar

[16]

M. Fuhrman, F. Masiero and G. Tessitore, Stochastic equations with delay: Optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations,, SIAM Journal on Control and Optimization, 48 (2010), 4624. doi: 10.1137/080730354. Google Scholar

[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), (2006), 133. doi: 10.1201/9781420028720.ch13. Google Scholar

[18]

V. B. Kolmanovskii and T. L. Maizenberg, Optimal control of stochastic systems with aftereffect,, Automation & Remote Control, 34 (1973), 39. Google Scholar

[19]

V. B. Kolmanovskii and L. E. Shaikhet, Control of Systems with Aftereffect,, Translation of Mathematical Monographs, (1996). Google Scholar

[20]

B. Larssen, Dynamic programming in stochastic control of systems with delay,, Stochastics & Stochastics Reports, 74 (2002), 651. doi: 10.1080/1045112021000060764. Google Scholar

[21]

B. Larssen and N. H. Risebro, When are HJB-equations in stochastic control of delay systems finite dimensional?,, Stochastic Analysis and Applications, 21 (2003), 643. doi: 10.1081/SAP-120020430. Google Scholar

[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), 295. doi: 10.1080/17442508.2011.652965. Google Scholar

[23]

S. E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications,, in Stochastic Analysis and Related Topics VI, (1996), 1. Google Scholar

[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, (2000). Google Scholar

[25]

B. Øksendal and A. Sulem, Maximum principles for optimal control of forward-backward stochastic differential equations with jumps,, SIAM Journal on Control and Optimization, 48 (2009), 2945. doi: 10.1137/080739781. Google Scholar

[26]

B. Øksendal and A. Sulem, Forward-backward stochastic differential games and stochastic control under model uncertainty,, Journal of Optimization Theory and Applications, 161 (2014), 22. doi: 10.1007/s10957-012-0166-7. Google Scholar

[27]

B. Øksendal, A. Sulem and T. S. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations,, Advances in Applied Probability, 43 (2011), 572. doi: 10.1239/aap/1308662493. Google Scholar

[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). doi: 10.1007/s10957-013-0484-4. Google Scholar

[29]

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation,, Systems & Control Letters, 14 (1990), 55. doi: 10.1016/0167-6911(90)90082-6. Google Scholar

[30]

S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations,, The Annals of Probability, 37 (2009), 877. doi: 10.1214/08-AOP423. Google Scholar

[31]

M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with stochastic differential utility,, Journal of Economic Theory, 89 (1999), 68. doi: 10.1006/jeth.1999.2558. Google Scholar

[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, (2011), 1210. Google Scholar

[33]

J. T. Shi and Z. Wu, Maximum principle for forward-backward stochastic control systems with random jumps and applications to finance,, Journal of Systems Science and Complexity, 23 (2010), 219. doi: 10.1007/s11424-010-7224-8. Google Scholar

[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), 151. doi: 10.1007/s00245-010-9115-8. Google Scholar

[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). Google Scholar

[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), 1230. doi: 10.1109/TAC.2009.2019794. Google Scholar

[37]

H. X. Wang and H. S. Zhang, LQ control for Itô type stochastic systems with input delays,, Automatica, 49 (2013), 3538. doi: 10.1016/j.automatica.2013.09.018. Google Scholar

[38]

J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

[39]

Z. Y. Yu, The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls,, Automatica, 48 (2012), 2420. doi: 10.1016/j.automatica.2012.06.082. Google Scholar

[40]

H. S. Zhang, G. R. Duan and L. H. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays,, Automatica, 42 (2006), 1465. doi: 10.1016/j.automatica.2006.04.007. Google Scholar

[41]

H. S. Zhang, G. Feng and C. Y. Han, Linear estimation for random delay systems,, Systems & Control Letters, 60 (2011), 450. doi: 10.1016/j.sysconle.2011.03.009. Google Scholar

[42]

X. Zhang, R. J. Elliott and T. K. Siu, A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance,, SIAM Journal on Control and Optimization, 50 (2012), 964. doi: 10.1137/110839357. Google Scholar

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