zbMATH — the first resource for mathematics

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.

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
Full Text: DOI