Kolmogorov Equations Driven by Cylindrical Stable Processes and Applications to Stochastic Optimal Control

柱型穩定過程所導出之半線性Kolmogorov方程式及其在隨機最佳化控制的應用

蘇國樑(Kuo-Liang Su)

Kolmogorov方程式 ； 隨機最佳化控制 ； 柱型α-穩定過程 ； Hamilton-Jacobi-Bellman方程式 ； kolmogorov equation ； stochastic optimal control ； cylindrical α-stable process ； hamilton-jacobi-bellman equation

中國統計學報

52卷3期（2014 / 09 / 01）

335 - 362

英文

本文解出柱型α-穩定過程所導出之半線性Kolmogorov方程式在無限維Hilbert空間上的溫和解；其次，解出最佳化控制問題所相關之Hamilton-Jacobi-Bellman方程式的解；然後，說明在隨機控制熱傳導方程式和波動方程式上的最佳化應用。

A mild solution of semilinear Kolmogorov equations driven by a cylindrical α-stable noise is solved in an infinite dimensional Hilbert space. The associated Hamilton-Jacobi-Bellman equation is also studied and solved for applications to the stochastic optimal control problems. It then shows applications to the controlled stochastic heat equation and wave equation.

1. Applebaum, D.(2009).Lévy processes and stochastic calculus.Cambridge:Cambridge University Press.
2. Bass, R. F.,Chen, Z.(2006).Systems of equations driven by stable processes.Probab. Theory Relat. Fields,134,175-214.
3. Bensoussan, A.,Da Prato, G.,Delfour, M. C.,Mitter, S. K.(2007).Representation and Control of Infinite Dimensional Systems.Birkhauser.
4. Cerrai, S.(2001).Second order PDE's in nite and in nite dimension, A probabilistic approach.Lecture Notes in Mathematics,1762
5. Da Prato, G.,Zabczyk, J.(2002).Second-Order Partial Differential Equations in Hilbert Spaces.Cambridge:Cambridge University Press.
6. Da Prato, G.,Zabczyk, J.(1996).Ergodicity for infinite-dimensional systems.Cambridge:Cambridge University Press.
7. Da Prato, G.,Zabczyk, J.(1992).Stochastic equations in infinite dimensions.Cambridge:Cambridge University Press.
8. Fuhrman, M.,Tessitore, G.(2002).Nonlinear Kolmogorov equations in infinite-dimensional spaces: the backward stochastic differential equations approach and applications to optimal control.Ann. Probab.,30(3),1397-1465.
9. Fuhrman, M.,Tessitore, G.(2004).Infinite horizon backward stochastic Differ-ential equations and elliptic equations in Hilbert space.Ann. Probab.,32(1B),607-660.
10. Goldys, B.,Gozzi, F.(2006).Second order parabolic Hamilton-Jacobi -Bellman equations in Hilbert spaces and stochastic control:L2µ approach.Stochastic Processes and their Applications,116,1932-1963.
11. Hu, Y.,Tessitore, G.(2007).BSDE on an infinite horizon and elliptic PDEs in infinite dimension.Nonlinear differ. equ. appl.,14,825-846.
12. Masiero, F.(2008).Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain.Stochastic Analysis and Applications,26(4),877-902.
13. Masiero, F.(2005).Semilinear Kolmogorov equations and applications to stochastic optimal control.Appl. Math. Optim,51(2),201-250.
14. Masiero, F.(2007).Regularizing properties for transition semigroups and semilinear parabolic equations in Banach spaces.Electron. J. Probab,12,387-419.
15. Peszat, E.,Zabczyk, J.(2007).Stochastic Partial Differential Equations with Levy noise.Cambridge:
16. Priola, E.,Xu, L.,Zabczyk, J.(2011).Exponential mixing for some SPDEs with Lévy noise.Stochastic and Dynamics,11,521-534.
17. Priola, E.,Zabczyk, J.(2003).Null Controllability with Vanishing Energy.SIAM J. Control Optim,42(3),1013-1032.
18. Priola, E.,Zabczyk, J.(2011).Structural properties of semilinear SPDEs driven by cylindrical stable processes.Probab. Theory Relat. Fields,149,97-137.
19. Royden, H. L.,Fitzpatrick, P. M.(2010).Real Analysis.Pearson Education, Inc. Prentice Hall.
20. Samorodnitsky, G.,Taqqu, M. S.(1994).Stable non-Gaussian random processes. Stochastic models with infinite variance.New York:Chapman and Hall.
21. Sato, K. I.(1999).Lévy processes and infinite divisible distributions.Cambridge:Cambridge University Press.
22. Su, K. L.(2010).A Note of Exponential Martingale in Banach Spaces.J. of Business,18,125-138.
23. Su, K. L.(2012).A Note on Applications of Infinite Dimensional Elliptic PDE to In nite Horizon Control Problems.J. of Business,20,257-274.
24. Su, K. L.(2011).A Note of Girsanov's Theorem in Banach Spaces.J. of Business,19,133-148.
25. Van Neerven, J. M. A. M.(2008).Stochastic Evolution Equation.Lecture Notes of the 11th Internet Seminar 2007/08