# American Institute of Mathematical Sciences

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 and 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), 2499-2522. doi: 10.1137/120882809. [2] N. Agram and B. Øksendal, Infinite horizon optimal control of forward-backward stochastic differential equations with delay, Journal of Computational and Applied Mathematics, Series B, 259 (2014), 336-349. 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), 471-492. 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), 233-249. 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), 604-619. 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), 1074-1080. 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), 1005-1026. 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 regime-switching diffusion model, Applied Mathematics and Optimization, 64 (2011), 155-169. doi: 10.1007/s00245-010-9130-9. [10] D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. 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), 1-71. doi: 10.1111/1467-9965.00022. [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), 664-693. 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), 69-89. doi: 10.1080/17442500008834259. [14] S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance & Stochastics, 15 (2011), 421-459. doi: 10.1007/s00780-010-0146-4. [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), 77-98. 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 Hamilton-Jacobi-Bellman equations, SIAM Journal on Control and Optimization, 48 (2010), 4624-4651. 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, 133-148. 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), 39-52. [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), 651-673. doi: 10.1080/1045112021000060764. [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), 643-671. doi: 10.1081/SAP-120020430. [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), 295-320. 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, 1-77. [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 forward-backward stochastic differential equations with jumps, SIAM Journal on Control and Optimization, 48 (2009), 2945-2976. doi: 10.1137/080739781. [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), 22-55. doi: 10.1007/s10957-012-0166-7. [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), 572-596. 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/s10957-013-0484-4. [29] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [30] S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations, The Annals of Probability, 37 (2009), 877-902. doi: 10.1214/08-AOP423. [31] M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with stochastic differential utility, Journal of Economic Theory, 89 (1999), 68-126. 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, 1210-1215. [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), 219-231. doi: 10.1007/s11424-010-7224-8. [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), 151-189. doi: 10.1007/s00245-010-9115-8. [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), 1230-1242. 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), 3538-3549. doi: 10.1016/j.automatica.2013.09.018. [38] J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3. [39] Z. Y. Yu, The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls, Automatica, 48 (2012), 2420-2432. doi: 10.1016/j.automatica.2012.06.082. [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), 1465-1476. 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), 450-459. 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 regime-switching jump-diffusion model and its application to finance, SIAM Journal on Control and Optimization, 50 (2012), 964-990. doi: 10.1137/110839357.

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##### 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), 2499-2522. doi: 10.1137/120882809. [2] N. Agram and B. Øksendal, Infinite horizon optimal control of forward-backward stochastic differential equations with delay, Journal of Computational and Applied Mathematics, Series B, 259 (2014), 336-349. 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), 471-492. 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), 233-249. 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), 604-619. 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), 1074-1080. 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), 1005-1026. 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 regime-switching diffusion model, Applied Mathematics and Optimization, 64 (2011), 155-169. doi: 10.1007/s00245-010-9130-9. [10] D. Duffie and L. G. Epstein, Stochastic differential utility, Econometrica, 60 (1992), 353-394. 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), 1-71. doi: 10.1111/1467-9965.00022. [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), 664-693. 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), 69-89. doi: 10.1080/17442500008834259. [14] S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance & Stochastics, 15 (2011), 421-459. doi: 10.1007/s00780-010-0146-4. [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), 77-98. 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 Hamilton-Jacobi-Bellman equations, SIAM Journal on Control and Optimization, 48 (2010), 4624-4651. 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, 133-148. 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), 39-52. [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), 651-673. doi: 10.1080/1045112021000060764. [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), 643-671. doi: 10.1081/SAP-120020430. [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), 295-320. 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, 1-77. [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 forward-backward stochastic differential equations with jumps, SIAM Journal on Control and Optimization, 48 (2009), 2945-2976. doi: 10.1137/080739781. [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), 22-55. doi: 10.1007/s10957-012-0166-7. [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), 572-596. 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/s10957-013-0484-4. [29] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [30] S. G. Peng and Z. Yang, Anticipated backward stochastic differential equations, The Annals of Probability, 37 (2009), 877-902. doi: 10.1214/08-AOP423. [31] M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with stochastic differential utility, Journal of Economic Theory, 89 (1999), 68-126. 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, 1210-1215. [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), 219-231. doi: 10.1007/s11424-010-7224-8. [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), 151-189. doi: 10.1007/s00245-010-9115-8. [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), 1230-1242. 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), 3538-3549. doi: 10.1016/j.automatica.2013.09.018. [38] J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3. [39] Z. Y. Yu, The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls, Automatica, 48 (2012), 2420-2432. doi: 10.1016/j.automatica.2012.06.082. [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), 1465-1476. 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), 450-459. 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 regime-switching jump-diffusion model and its application to finance, SIAM Journal on Control and Optimization, 50 (2012), 964-990. doi: 10.1137/110839357.
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