ON ERGODIC BELLMAN EQUATION

The talk is devoted to the results on ergodic Bellman's equation for the case of a uniformly ergodic controlled diffusion with variable diffusion and drift coefficients both depending on control; the diffusion coefficient must be a scalar function. The convergence of Howard's iterative reward improvement algorithm to the unique solution of Bellman's equation is established. Models of controlled diffusion with switching can also be considered. Solutions of corresponding stochastic differential equations may be weak (which is technically easier) or strong (which requires certain moderate additional efforts). For a first acquaintance with the simplest 1d model see the preprint https://doi.org/10.48550/arXiv.1812.10665. About the beginning of stochastic calculus see https://disk.yandex.ru/i/r-eW_8GD2hAYnA, and for more details about it see, e.g., https://teach-in.ru/course/theory-of-stochastic-differential-equations. It is advised to recall Ito’s formula, although, an effort will be done to minimise the “stochastic” counterpart of the talk. On the other hand, a short introduction to the latter area may be offered, by a seminar participants request.