Providing an introduction to stochastic optimal control in inï¬nite dimension, this book gives a complete account of the theory of second-order HJB equations in inï¬nite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in inï¬nite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs,and in PDEs in inï¬nite dimension. Readers from other ï¬elds who want to learn the basic theory will also ï¬nd it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to stochastic optimal control problems in ï¬nite dimension, and the basics of stochastic analysis and stochastic equations in inï¬nite-dimensional spaces.
