  In this paper we study a family of nonlinear (conditional) expectations that
can be understood as a semimartingale with uncertain local characteristics.
Here, the differential characteristics are prescribed by a time and
path-dependent set-valued function. We show that the associated control problem
coincides with both its weak and relaxed counterparts. Furthermore, we
establish regularity properties of the value function and discuss their
relation to Feller properties of nonlinear semigroups. In the Markovian case we
provide conditions that allow us to identify the corresponding semigroup as the
unique viscosity solution to a nonlinear Hamilton-Jacobi-Bellman equation. To
illustrate our results we discuss a random $G$-double exponential L\'evy
setting.
