Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

Meijiao Wang; Qiuhong Shi; Maoning Tang; Qingxin Meng; Meijiao Wang; Qiuhong Shi; Maoning Tang; Qingxin Meng

doi:10.3934/math.2022137

AIMS Mathematics

2022, Volume 7, Issue 2: 2427-2455. doi: 10.3934/math.2022137

Previous Article Next Article

Research article

Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

1.
Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
Department of Mathematics, Huzhou University, Zhejiang 313000, China

Received: 12 April 2021 Accepted: 04 November 2021 Published: 12 November 2021
MSC : 60H10, 93E24

The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens ^[21]). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.
- stochastic evolution equation,
- Teugels martingales,
- optimal control,
- stochastic maximum principle,
- verification theorem
Citation: Meijiao Wang, Qiuhong Shi, Maoning Tang, Qingxin Meng. Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes[J]. AIMS Mathematics, 2022, 7(2): 2427-2455. doi: 10.3934/math.2022137

Related Papers:

Abstract

The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens ^[21]). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.

References

[1]	S. Albeverio, J. L. Wu, T. S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Processes Appl., 74 (1998), 21-36. doi: 10.1016/S0304-4149(97)00112-9. doi: 10.1016/S0304-4149(97)00112-9
[2]	K. Bahlali, M. Eddahbi, E. Essaky, BSDE associated with Lévy processes and application to PDIE, J. Appl. Math. Stochastic Anal., 16 (2003), 1-17. doi: 10.1155/s1048953303000017. doi: 10.1155/s1048953303000017
[3]	A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst., 315 (1983), 387-406. doi: 10.1016/0016-0032(83)90059-5. doi: 10.1016/0016-0032(83)90059-5
[4]	P. Benner, C. Trautwein, A linear quadratic control problem for the stochastic heat equation driven by Q-Wiener processes, J. Math. Anal. Appl., 457 (2018), 776-802. doi: 10.1016/j.jmaa.2017.08.052. doi: 10.1016/j.jmaa.2017.08.052
[5]	S. Chen, S. Tang, Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process, arXiv. Available from: https://arXiv.org/abs/1007.3201.
[6]	P. L. Chow, Stochastic partial differential equations, CRC Press, 2014. doi: 10.1201/9781420010305.
[7]	G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, 2014. doi: 10.1017/CBO9781107295513.
[8]	K. Du, Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362. doi: 10.1137/120882433. doi: 10.1137/120882433
[9]	I. Ekeland, R. Témam, Convex analysis and variational problems, North-Holland, Amsterdam, 1999. doi: 10.1137/1.9781611971088.
[10]	M. El Otmani, Generalized BSDE driven by a Lévy process, Int. J. Stochastic Anal., 2006 (2006), 085407. doi: 10.1155/JAMSA/2006/85407. doi: 10.1155/JAMSA/2006/85407
[11]	M. El Otmani, Backward stochastic differential equations associated with Lévy processes and partial integro-differential equations, Commun. Stochastic Anal., 2 (2008), 277-288. doi: 10.31390/cosa.2.2.07. doi: 10.31390/cosa.2.2.07
[12]	M. Fuhrman, Y. Hu, G. Tessitor, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217. doi: 10.1016/j.crma.2012.07.009. doi: 10.1016/j.crma.2012.07.009
[13]	M. Fuhrman, C. Orrieri, Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift, SIAM J. Control Optim., 54 (2016), 341-371. doi: 10.1137/15m1012888. doi: 10.1137/15m1012888
[14]	I. Gyöngy, N. V. Krylov, On stochastics equations with respect to semimartingales ii. itô formula in banach spaces, Stochastics, 6 (1982), 153-173. doi: 10.1080/17442508208833202. doi: 10.1080/17442508208833202
[15]	Y. Hu, N-person differential games governed by semilinear stochastic evolution systems, Appl. Math. Optim., 24 (1991), 257-271. doi: 10.1007/BF01447745. doi: 10.1007/BF01447745
[16]	Y. Hu, S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastic Rep., 33 (1990), 159-180. doi: 10.1080/17442509008833671. doi: 10.1080/17442509008833671
[17]	S. Lenhart, J. Xiong, J. Yong, Optimal controls for stochastic partial differential equations with an application in population modeling, SIAM J. Control Optim., 54 (2016), 495-535. doi: 10.1137/15m1010233. doi: 10.1137/15m1010233
[18]	Q. Lü, X. Zhang, General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions, Springer, 2014. doi: 10.1007/978-3-319-06632-5.
[19]	Q. Meng, M. Tang, Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes, Sci. China Ser. F-Inf. Sci., 52 (2009), 1982. doi: 10.1007/s11432-009-0191-9. doi: 10.1007/s11432-009-0191-9
[20]	K. Mitsui, Y. Tabata, A stochastic linear-quadratic problem with Lévy processes and its application to finance, Stochastic Processes Appl., 118 (2008), 120-152. doi: 10.1016/j.spa.2007.03.011. doi: 10.1016/j.spa.2007.03.011
[21]	D. Nualart, W. Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Processes Appl., 90 (2000), 109-122. doi: 10.1016/s0304-4149(00)00035-1. doi: 10.1016/s0304-4149(00)00035-1
[22]	D. Nualart, W. Schoutens, Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance, Bernoulli, 7 (2001), 761-776. doi: 10.2307/3318541. doi: 10.2307/3318541
[23]	B. Øksendal, F. Prosk, T. Zhang, Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields, Stochastics Int. J. Probab. Stochastic Processes, 77 (2005), 381-399. doi: 10.1080/17442500500213797. doi: 10.1080/17442500500213797
[24]	C. Orrieri, P. Veverka, Necessary stochastic maximum principle for dissipative systems on infinite time horizon, ESAIM: Control Optim. Calculus Var., 23 (2017), 337-371. doi: 10.1051/cocv/2015054. doi: 10.1051/cocv/2015054
[25]	S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054. doi: 10.1137/0328054
[26]	C. Prévôt, M. Röckner, A concise course on stochastic partial differential equations, Berlin: Springer, 2007. doi: 10.1007/978-3-540-70781-3.
[27]	Y. Ren, H. Dai, R. Sakthivel, Approximate controllability of stochastic differential systems driven by a Lévy process, Int. J. Control, 86 (2013), 1158-1164. doi: 10.1080/00207179.2013.786188. doi: 10.1080/00207179.2013.786188
[28]	Y. Ren, M. El Otmani, Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition, J. Comput. Appl. Math., 233 (2010), 2027-2043. doi: 10.1016/j.cam.2009.09.037. doi: 10.1016/j.cam.2009.09.037
[29]	Y. Ren, X. Fan, Reflected backward stochastic differential equations driven by a Lévy process, ANZIAM J., 50 (2009), 486-500. doi: 10.1017/s1446181109000303. doi: 10.1017/s1446181109000303
[30]	M. Röckner, T. Zhang, Stochastic evolution equations of jump type: Existenc uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279. doi: 10.1007/s11118-006-9035-z. doi: 10.1007/s11118-006-9035-z
[31]	Y. Ren, R. Sakthivel, Existenc uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, J. Math. Phys., 53 (2012), 073517. doi: 10.1063/1.4739406. doi: 10.1063/1.4739406
[32]	R. Sakthivel, Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4517-4523. doi: 10.1016/j.cnsns.2012.04.020. doi: 10.1016/j.cnsns.2012.04.020
[33]	H. Tang, Z. Wu, Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes, J. Syst. Sci. Complex., 22 (2009), 122-136. doi: 10.1007/s11424-009-9151-0. doi: 10.1007/s11424-009-9151-0
[34]	M. Tang, Q. Meng, Stochastic evolution equations of Jump type with random coefficients: Existence, uniqueness and optimal control, Sci. China Inf. Sci., 60 (2017), doi: 10.1007/s11432-016-9107-1. doi: 10.1007/s11432-016-9107-1
[35]	M. Tang, Q. Zhang, Optimal variational principle for backward stochastic control systems associated with Lévy processes, Sci. China Math., 55 (2012), 745-761. doi: 10.1007/s11425-012-4370-6. doi: 10.1007/s11425-012-4370-6
[36]	J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), 1-43. doi: 10.1007/s11118-004-2950-y. doi: 10.1007/s11118-004-2950-y
[37]	W. A. Woyczyński, Lévy processes in the physical sciences, In: O. E. Barndorff-Nielsen, S. I. Resnick, T. Mikosch, Lévy processes, Boston: Birkhäuser, (2001), 241-266. doi: 10.1007/978-1-4612-0197-7_11.
[38]	X. Yang, J. Zhai, T. Zhang, Large deviations for SPDEs of jump type, Stoch. Dynam., 15 (2015), 1550026. doi: 10.1142/s0219493715500264. doi: 10.1142/s0219493715500264
[39]	H. Zhao, S. Xu, Freidlin-Wentzell's large deviations for stochastic evolution equations with Poisson jumps, Adv. Pure Math., 6 (2016), 676-694. doi: 10.4236/apm.2016.610056. doi: 10.4236/apm.2016.610056
[40]	J. Zhai, T. Zhang, Large deviations for 2-D stochastic Navier-tokes equations driven by multiplicative Lévy noises, Bernoulli, 21 (2015), 2351-2392. doi: 10.3150/14-BEJ647. doi: 10.3150/14-BEJ647
[41]	X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478. doi: 10.1137/0331068. doi: 10.1137/0331068

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)