  In this paper we study a family of nonlinear (conditional) expectations that
can be understood as a continuous semimartingale with uncertain local
characteristics. Here, the differential characteristics are prescribed by a
set-valued function that depends on time and path in a non-Markovian way. We
provide a dynamic programming principle for the nonlinear expectation and we
link the corresponding value function to a variational form of a nonlinear
path-dependent partial differential equation. In particular, we establish
conditions that allow us to identify the value function as the unique viscosity
solution. Furthermore, we prove that the nonlinear expectation solves a
nonlinear martingale problem, which confirms our interpretation as a nonlinear
semimartingale.
