Abstract
The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon. Supposing that some sufficiently smooth conditions hold, by the dynamic programming principle, the author builds the Hamilton-Jacobi-Bellman (HJB for short) variational inequality for the value function. The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variational inequality.
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References
Bardi, M. and Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, MA, 1997.
Bensoussan, A., Stochastic Control by Functional Analysis Methods, North-Holland Publishing Co., Amsterdam, New York, 1982.
Bensoussan, A. and Lions, J. L., Applications of Variational Inequalities in Stochastic Control, North-Holland Publishing Co., Amsterdam, New York, 1982.
Chang, M. H., Pang, T. and Yong, J. M., Optimal stopping problem for stochastic differntial equations with random coefficients, SIAM J. Control OPtim., 48(2), 2009, 941–971.
Crandall, M. G., Ishii, H. and Lions, P. L., User’s guide to the viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27(1), 1992, 1–67.
Crandall, M. G. and Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277(1), 1983, 1–42.
El Karoui, N., Les Aspects Probabilistes du Contrôle Stochastique, Lecture Notes in Mathematics, 876, Springer-Verlag, Berlin, 1981.
Fleming, W. H. and Soner, H. M., Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.
Friedman, A., Optimal stopping problems in stochastic control, SIAM Rev., 21(1), 1979, 71–80.
Friedman, A., Variational Principle and Free-Boundary Problems, Wiley, New York, 1982.
Friedman, A., Optimal control for variational inequalities, SIAM J. Control Optim., 24(3), 1986, 439–451.
Friedman, A., Optimal control for parabolic variational inequalities, SIAM J. Control Optim., 25(2), 1987, 482–497.
Friedman, A., Huang, S. and Yong, J., Bang-bang optimal control for the dam problem, Appl. Math. Optim., 15(1), 1987, 65–85.
Friedman, A., Huang, S. and Yong, J., Optimal periodic control for the two-phase stefan problem, SIAM J. Control Optim., 26(1), 1988, 23–41.
Goreac, D. and Serea, O., Mayer and optimal stopping stochastic control problems with discontinuous cost, J. Math. Anal. and Appl., 380(1), 2011, 327–342.
Karatzas, I. and Shreve, S. E., Brownian Motion and Stochastic Calculus, 2nd ed., Springer-Verlag, Berlin, New York, 1991.
Karatzas, I. and Shreve, S. E., Methods of Mathematical Finance, Springer-Verlag, New York, 1998.
Karatzas, I. and Zamfirescu, I., Martingale approach to stochastic control with discretionary stopping, Appl. Math. Optim., 53(2), 2006, 163–184.
Karatzas, I. and Zamfirescu, I., Martingale approach to stochastic differntial games of control and stopping, The Annals of Probability, 36(4), 2008, 1495–1527.
Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Application, Academic Press, New York, 1980.
Koike, S. and Morimoto, H., Varitional inequalities for leavable bouned-velocity control, Appl. Math. Optim., 48(1), 2003, 1–20.
Koike, S., Morimoto, H. and Sakguchi, S., A linear-quadratic control problem with discretionary stopping, Discrte Contin. Dyn. Syst. Ser. B, 8(2), 2007, 261–277.
Krylov, N. V., Controlled Diffusion Processes, Springer-Verlag, New York, 1980.
Morimoto, H., Variational inequalities for combined control and stopping, SIAM J. Control Optim., 42(2), 2003, 686–708.
Øksendal, B., Stochastic Differential Equations, 5th ed., Springer-Verlag, Berlin, 1998.
Pham, H., Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems, Estimates and Control, 8(1), 1998, 1–27.
Shrayev, A. N., Optimal Stopping Rules, Springer-Verlag, Berlin, 1978.
Yong, J. M. and Zhou, X. Y., Stochastic Controls: Hamiltonian Systems and HJB Equations, SpringerVerlag, New York, 1999.
Acknowledgement
The author deeply thanks Professor Shanjian Tang for his very useful help and encouragement. This work is part of the author’s Ph.D thesis at Fudan University.
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Hu, S. Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control. Chin. Ann. Math. Ser. B 41, 793–820 (2020). https://doi.org/10.1007/s11401-020-0234-5
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DOI: https://doi.org/10.1007/s11401-020-0234-5