Ergodic Control for Lévy-Driven Linear Stochastic Equations in Hilbert Spaces
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In this paper, controlled linear stochastic evolution equations driven by square integrable Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied.
KeywordsLévy-driven SPDEs Ergodic control Controlled Lévy driven equations
Mathematics Subject Classification60H15 93E20
The authors are grateful to Szymon Peszat for his valuable comments and remarks. K. Kadlec was supported by the Charles University in Prague, Project GA UK No. 322715 and SVV-2015-260225. B. Maslowski was supported by Grant GACR No. 15-088195.
- 8.Duncan, T. E., Maslowski, B., and Pasik-Duncan, B.: Ergodic control of linear stochastic equations in a Hilbert space with fractional Brownian motions. To appear in Banach Center Publications, vol. 105Google Scholar
- 14.Lipster, RSh, Shiryayev, A.N.: Theory of Martingales. Kluwer Academic Publishers, Dobrecht (1989)Google Scholar