The maximum principle for discounted optimal control of partially observed forward-backward stochastic systems with jumps on infinite horizon. (English) Zbl 1521.93212
Summary: This paper is concerned with a discounted optimal control problem of partially observed forward-backward stochastic systems with jumps on infinite horizon. The control domain is convex and a kind of infinite horizon observation equation is introduced. The uniquely solvability of infinite horizon forward (backward) stochastic differential equation with jumps is obtained and more extended analyses, especially for the backward case, are made. Some new estimates are first given and proved for the critical variational inequality. Then a maximum principle is obtained by introducing some infinite horizon adjoint equations whose uniquely solvabilities are guaranteed necessarily. Finally, some comparisons are made with two kinds of representative infinite horizon stochastic systems and their related optimal controls.
MSC:
| 93E20 | Optimal stochastic control |
| 49N10 | Linear-quadratic optimal control problems |
| 49N70 | Differential games and control |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
partially observed maximum principle; infinite horizon forward-backward stochastic differential equation with jumps; discounted optimal controlReferences:
| [1] | N. Agram, S. Haadem, B. Øksendal and F. Proske, A maximum principle for infinite horizon delay equations. SIAM J. Math. Anal. 45 (2013) 2499-2522. · Zbl 1273.93175 · doi:10.1137/120882809 |
| [2] | N. Agram and B. Øksendal, Infinite horizon optimal control of forward-backward stochastic differential equations with delay. J. Comput. Appl. Math. 259 (2014) 336-349. · Zbl 1320.60121 · doi:10.1016/j.cam.2013.04.048 |
| [3] | S.M. Aseev and A.V. Kryazhimskii, The Pontryagin maximum principle and optimal economic growth problems. Proc. Steklov Inst. Math. 257 (2007) 1-255. · Zbl 1215.49001 · doi:10.1134/S0081543807020010 |
| [4] | S.M. Aseev and V.M. Veliov, Another view of the maximum principle for infinite-horizon optimal control problems in economics. Russ. Math. Surv. 74 (2019) 963-1011. · Zbl 1480.49022 · doi:10.1070/RM9915 |
| [5] | P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, L^p solutions of backward stochastic differential equations. Stochastic Process. Appl. 108 (2003) 109-129. · Zbl 1075.65503 · doi:10.1016/S0304-4149(03)00089-9 |
| [6] | Z.J. Chen, Existence and uniqueness for BSDE with stopping time. Chin. Sci. Bull. 43 (1998) 96-99. · Zbl 0994.60063 · doi:10.1007/BF02883914 |
| [7] | F. Confortola, L^p solution of backward stochastic differential equations driven by a marked point process. Math. Control Signal Syst. 31 (2019) 1. · Zbl 1405.93230 · doi:10.1007/s00498-018-0230-4 |
| [8] | R.W.R. Darling and E. Pardoux, Backward SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25 (1997) 1135-1159. · Zbl 0895.60067 |
| [9] | D. Duffie and L.G. Epstein, Stochastic differential utility. Econometrica 60 (1992) 353-394. · Zbl 0763.90005 |
| [10] | S. Haadem, B. Øksendal and F. Proske, Maximum principles for jump diffusion processes with infinite horizon. Automatica J. IFAC 49 (2013) 2267-2275. · Zbl 1364.93861 · doi:10.1016/j.automatica.2013.04.011 |
| [11] | H. Halkin, Necessary conditions for optimal control problems with infinite horizons. Econometrica 42 (1974) 267-272. · Zbl 0301.90009 |
| [12] | T. Hao and Q.X. Meng, A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps. ESAIM: Control, Optim. Calcu. Varia. 26 (2020) 87. · Zbl 1460.93109 · doi:10.1051/cocv/2020008 |
| [13] | M.S. Hu, Stochastic global maximum principle for optimization with recursive utilities. Probab. Uncertain. Quant. Risk 2 (2017) 1-20. · Zbl 1432.93380 · doi:10.1186/s41546-017-0014-7 |
| [14] | M.S. Hu, S.L. Ji and X.L. Xue, A global stochastic maximum principle for full coupled forward-backward stochastic systems. SIAM J. Control Optim. 56 (2018) 4309-4335. · Zbl 1403.93197 · doi:10.1137/18M1179547 |
| [15] | J.P. Huang, J.M. Yong and H.C. Zhou, Infinite horizon linear quadratic overtaking optimal control problems. SIAM J. Control Optim. 59 (2021) 1312-1340. · Zbl 1461.49044 · doi:10.1137/20M1361675 |
| [16] | N. El Karoui, S.G. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1-71. · Zbl 0884.90035 |
| [17] | X.J. Li and S.J. Tang, General necessary conditions for partially observed optimal stochastic controls. J. Appl. Probab. 32 (1995) 1118-1137. · Zbl 0844.93076 · doi:10.2307/3215225 |
| [18] | H.P. Ma and B. Liu, Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information. Eur. J. Control 36 (2017) 43-50. · Zbl 1506.93102 · doi:10.1016/j.ejcon.2017.04.001 |
| [19] | B. Maslowski and P. Veverka, Sufficient stochastic maximum principle for discounted control problem. Appl. Math. Optim. 70 (2014) 225-252. · Zbl 1303.93189 · doi:10.1007/s00245-014-9241-9 |
| [20] | H.W. Mei, Q.M. Wei and J.M. Yong, Optimal ergodic control of linear stochastic differential equations with quadratic cost functionals having indefinite weights. SIAM J. Control Optim. 59 (2021) 584-613. · Zbl 1458.93267 · doi:10.1137/20M1334206 |
| [21] | P. Muthukumar and R. Deepa, Infinite horizon optimal control of forward-backward stochastic system driven by Teugels martingales with Lévy processes. Stoch. Dyn. 17 (2017) 1750020. · Zbl 1360.49002 · doi:10.1142/S0219493717500204 |
| [22] | H. Nagai, Risk-sensitive dynamic asset management with partial information. Stochastics in Finite and Infinite Dimension. Trends. Math. Birkhäuser, Boston (2000) 321-339. · Zbl 1013.91057 |
| [23] | H. Nagai and S.G. Peng, Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann. Appl. Probab. 12 (2002) 173-195. · Zbl 1042.91048 · doi:10.1214/aoap/1015961160 |
| [24] | C. Orrieri, G. Tessitore and P. Veverka, Ergodic maximum principle for stochastic systems. Appl. Math. Optim. 79 (2019) 567-591. · Zbl 1427.60131 · doi:10.1007/s00245-017-9448-7 |
| [25] | C. Orrieri and P. Veverka, Necessary stochastic maximum principle for dissipative systems on infinite time horizon. ESAIM: Control, Optim. Calcu. Varia. 23 (2017) 337-371. · Zbl 1354.93178 · doi:10.1051/cocv/2015054 |
| [26] | E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, in edited by F.H. Clark and R.J. Stern. Nonlinear Analysis, Differential Equations and Control. Kluwer Academic, Dordrecht (1999) 503-549. · Zbl 0959.60049 |
| [27] | S.G. Peng, A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28 (1990) 966-979. · Zbl 0712.93067 · doi:10.1137/0328054 |
| [28] | S.G. Peng, Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim. 27 (1993) 125-144. · Zbl 0769.60054 · doi:10.1007/BF01195978 |
| [29] | S.G. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37 (1991) 61-74. · doi:10.1080/17442509108833727 |
| [30] | S.G. Peng and Y.F. Shi, Infinite horizon forward-backward stochastic differential equations. Stoch. Process. Appl. 85 (2000) 75-92. · Zbl 0997.60062 · doi:10.1016/S0304-4149(99)00066-6 |
| [31] | M.C. Quenez and A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123 (2013) 3328-3357. · Zbl 1285.93091 · doi:10.1016/j.spa.2013.02.016 |
| [32] | S. P. Sethi, Optimal Control Theory. Applications to Management Science and Economics, 4rd ed. Springer, Cham (2019). · Zbl 1412.49001 · doi:10.1007/978-3-319-98237-3 |
| [33] | K. Shell, Applications of Pontryagin’s maximum principle to economics, in Mathematical systems theory and economics (Varenna 1967). Lect. Notes Oper. Res. Math. Econom. Springer, Berlin 11 (1969) 241-292. · Zbl 0177.23403 |
| [34] | Y.F. Shi and H.Z. Zhao, Forward-backward stochastic differential equations on infinite horizon and quasilinear elliptic PDEs. J. Math. Anal. Appl. 485 (2020) 123791. · Zbl 1431.60049 · doi:10.1016/j.jmaa.2019.123791 |
| [35] | R. SiTu, A maximum principle for optimal controls of stochastic systems with random jumps, in Proceedings of National Conference on Control Theory and Applications, Qingdao (1991). |
| [36] | V.K. Socgnia and O.M. Pamen, An infinite horizon stochastic maximum principle for discounted control problem with Lipschitz coefficients. J. Math. Anal. Appl. 422 (2015) 684-711. · Zbl 1341.49031 · doi:10.1016/j.jmaa.2014.09.010 |
| [37] | Y.Z. Song, S.J. Tang and Z. Wu, The maximum principle for progressive optimal stochastic control problems with random jumps. SIAM J. Control Optim. 58 (2020) 2171-2187. · Zbl 1447.93378 · doi:10.1137/19M1292308 |
| [38] | S.J. Tang and X.J. Li, Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 (1994) 1447-1475. · doi:10.1137/S0363012992233858 |
| [39] | C.C. von Weizsäcker, Existence of optimal programs of accumulation for an infinite time horizon. Rev. Econ. Stud. 32 (1965) 85-104. · doi:10.2307/2296054 |
| [40] | G.C. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control 54 (2009) 1230-1242. · Zbl 1367.93725 · doi:10.1109/TAC.2009.2019794 |
| [41] | G.C. Wang, Z. Wu and J. Xiong, Maximum principles for forward-backward stochastic control systems with correlated state and observation noises. SIAM J. Control Optim. 51 (2013) 491-524. · Zbl 1262.93027 · doi:10.1137/110846920 |
| [42] | Q.M. Wei and Z.Y. Yu, Infinite horizon forward-backward SDEs and open-loop optimal controls for stochastic linear-quadratic problems with random coefficients. SIAM J. Control Optim. 59 (2021) 2594-2623. · Zbl 1467.93340 · doi:10.1137/20M1360517 |
| [43] | Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems. Sci. China Inf. Sci. 53 (2010) 2205-2214. · Zbl 1227.93116 · doi:10.1007/s11432-010-4094-6 |
| [44] | Z. Wu, Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration. J. Aust. Math. Soc. 74 (2003) 249-266. · Zbl 1029.60047 · doi:10.1017/S1446788700003281 |
| [45] | B.X. Yang and J.B. Wu, Infinite horizon optimal control for mean-field stochastic delay systems driven by Teugels martingales under partial information. Optim. Control Appl. Methods 41 (2020) 1371-1397. · Zbl 1469.93124 · doi:10.1002/oca.2602 |
| [46] | J.L. Yin, Forward-backward SDEs with random terminal time and applications to pricing special European-type options for a large investor. Bull. Sci. Math. 135 (2011) 883-895. · Zbl 1230.60064 · doi:10.1016/j.bulsci.2011.04.001 |
| [47] | J.L. Yin, On solutions of a class of infinite horizon FBSDEs. Stat. Probab. Lett. 78 (2008) 2412-2419. · Zbl 1151.60031 · doi:10.1016/j.spl.2008.03.002 |
| [48] | J.L. Yin and X.R. Mao, The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications. J. Math. Anal. Appl. 346 (2008) 345-358. · Zbl 1157.60061 · doi:10.1016/j.jmaa.2008.05.072 |
| [49] | J.L. Yin, R. SiTu, On solutions of forward-backward stochastic differential equations with Poisson jumps. Stoch. Anal. Appl. 21 (2003) 1419-1448. · Zbl 1037.60062 · doi:10.1081/SAP-120026113 |
| [50] | Z.Y. Yu, Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems. ESAIM: Control, Optim. Calc. Var. 23 (2017) 1331-1359. · Zbl 1375.60104 · doi:10.1051/cocv/2016055 |
| [51] | S.Q. Zhang, J. Xiong and X.D. Liu, Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching. Sci. China Inf. Sci. 61 (2018) 070211. · doi:10.1007/s11432-017-9267-0 |
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