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A note on general Tauberian-type results for controlled stochastic dynamics. (English) Zbl 1347.60115
Summary: In this note, we show that, in the context of stochastic control systems, the uniform existence of a limit of CesĂ ro averages implies the existence of uniform limits for averages with respect to a wide class of measures dominated by the Lebesgue measure and satisfying some asymptotic condition. It gives a partial answer to the problem mentioned in [J. Renault, J. Dyn. Games 1, No. 3, 471–484 (2014; Zbl 1305.90404)] and it provides an alternative method for the approach in [X. Li et al., Discrete Contin. Dyn. Syst. 36, No. 4, 2113–2132 (2016; Zbl 1326.93129)] (in the deterministic control setting). Finally, we mention that the arguments rely essentially on integration-by-parts and are applicable to general deterministic or stochastic control problems.

MSC:
60J60 Diffusion processes
60J75 Jump processes (MSC2010)
60F99 Limit theorems in probability theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J25 Continuous-time Markov processes on general state spaces
60G57 Random measures
49J55 Existence of optimal solutions to problems involving randomness
93E20 Optimal stochastic control
93E15 Stochastic stability in control theory
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