American Institute of Mathematical Sciences

November  2015, 35(11): 5499-5519. doi: 10.3934/dcds.2015.35.5499

A stochastic maximum principle with dissipativity conditions

 1 Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1. 27100, Pavia, Italy

Received  September 2013 Revised  October 2014 Published  May 2015

In this paper we prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a finite dimensional stochastic differential equation, driven by a multidimensional Wiener process. We drop the usual Lipschitz assumption on the drift term and substitute it with dissipativity conditions, allowing polynomial growth. The control enters both the drift and the diffusion term and takes values in a general metric space.
Citation: Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
References:
 [1] K. Bahlali, B. Mezerdi and Y. Ouknine, The maximum principle for optimal control of diffusions with non-smooth coefficients, Stochastics Stochastics Rep, 57 (1996), 303-316. doi: 10.1080/17442509608834065. [2] A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst, 315 (1983), 387-406. doi: 10.1016/0016-0032(83)90059-5. [3] P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations, Stochastic Process. Appl., 108 (2001), 604-618. [4] P. Briand and F. Confortola, Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators, Appl. Math. Optim., 57 (2008), 149-176. doi: 10.1007/s00245-007-9014-9. [5] A. Cadenillas and I. Karatzas, The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33 (1995), 590-624. doi: 10.1137/S0363012992240722. [6] G. Da Prato, M. Iannelli and L. Tubaro, Dissipative functions and finite-dimensional stochastic differential equations, J. Math. Pures Appl., 57 (1978), 173-180. [7] K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362. doi: 10.1137/120882433. [8] K. Du and Q. Meng, Stochastic maximum principle for infinite dimensional control systems, preprint, arXiv:1208.0529. [9] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, C. R. Math. Acad. Sci. Paris, 350 (2012), 683-688. doi: 10.1016/j.crma.2012.07.009. [10] Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180. doi: 10.1080/17442509008833671. [11] Q. Lü and X. Zhang, General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions, preprint, arXiv:1204.3275. [12] N. I. Mahmudov, General necessary conditions of optimality for stochastic systems with controllable diffusion, (in Russian) Proc. Workshop Statistics and Control of Random Processes, (1989), 135-138. [13] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, $2^{nd}$ edition, Universitext, Springer, Berlin, 2007. doi: 10.1007/978-3-540-69826-5. [14] E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear Analysis, Differential Equations and Control, 528 (1999), 503-549. [15] S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054. [16] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74. doi: 10.1080/17442509108833727. [17] S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475. doi: 10.1137/S0363012992233858. [18] S. Tang and X. Li, Maximum principle for optimal control of distributed parameter stochastic systems with random jumps, Lecture Notes in Pure and Applied Mathematics, 152 (1994), 867-890. [19] J. 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. [20] X. Y. Zhou, A unified treatment of maximum principle and dynamic programming in stochastic controls, Stochastics Stochastics Rep., 36 (1991), 137-161. doi: 10.1080/17442509108833715.

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References:
 [1] K. Bahlali, B. Mezerdi and Y. Ouknine, The maximum principle for optimal control of diffusions with non-smooth coefficients, Stochastics Stochastics Rep, 57 (1996), 303-316. doi: 10.1080/17442509608834065. [2] A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst, 315 (1983), 387-406. doi: 10.1016/0016-0032(83)90059-5. [3] P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations, Stochastic Process. Appl., 108 (2001), 604-618. [4] P. Briand and F. Confortola, Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators, Appl. Math. Optim., 57 (2008), 149-176. doi: 10.1007/s00245-007-9014-9. [5] A. Cadenillas and I. Karatzas, The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33 (1995), 590-624. doi: 10.1137/S0363012992240722. [6] G. Da Prato, M. Iannelli and L. Tubaro, Dissipative functions and finite-dimensional stochastic differential equations, J. Math. Pures Appl., 57 (1978), 173-180. [7] K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362. doi: 10.1137/120882433. [8] K. Du and Q. Meng, Stochastic maximum principle for infinite dimensional control systems, preprint, arXiv:1208.0529. [9] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, C. R. Math. Acad. Sci. Paris, 350 (2012), 683-688. doi: 10.1016/j.crma.2012.07.009. [10] Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180. doi: 10.1080/17442509008833671. [11] Q. Lü and X. Zhang, General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions, preprint, arXiv:1204.3275. [12] N. I. Mahmudov, General necessary conditions of optimality for stochastic systems with controllable diffusion, (in Russian) Proc. Workshop Statistics and Control of Random Processes, (1989), 135-138. [13] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, $2^{nd}$ edition, Universitext, Springer, Berlin, 2007. doi: 10.1007/978-3-540-69826-5. [14] E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear Analysis, Differential Equations and Control, 528 (1999), 503-549. [15] S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054. [16] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74. doi: 10.1080/17442509108833727. [17] S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475. doi: 10.1137/S0363012992233858. [18] S. Tang and X. Li, Maximum principle for optimal control of distributed parameter stochastic systems with random jumps, Lecture Notes in Pure and Applied Mathematics, 152 (1994), 867-890. [19] J. 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. [20] X. Y. Zhou, A unified treatment of maximum principle and dynamic programming in stochastic controls, Stochastics Stochastics Rep., 36 (1991), 137-161. doi: 10.1080/17442509108833715.
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