  In this paper we study stochastic control problems with delayed information,
that is, the control at time $t$ can depend only on the information observed
before time $t-H$ for some delay parameter $H$. Such delay occurs frequently in
practice and can be viewed as a special case of partial observation. When the
time duration $T$ is smaller than $H$, the problem becomes a deterministic
control problem in stochastic setting. While seemingly simple, the problem
involves certain time inconsistency issue, and the value function naturally
relies on the distribution of the state process and thus is a solution to a
nonlinear master equation. Consequently, the optimal state process solves a
McKean-Vlasov SDE. In the general case that $T$ is larger than $H$, the master
equation becomes path-dependent and the corresponding McKean-Vlasov SDE
involves the conditional distribution of the state process. We shall build
these connections rigorously, and obtain the existence of classical solution of
these nonlinear (path-dependent) master equations in a special case.

Probability