  In this paper we study a family of nonlinear (conditional) expectations that
can be understood as a stochastic process with uncertain parameters. We develop
a general framework which can be seen as a version of the martingale problem
method of Stroock and Varadhan with parameter uncertainty. To illustrate our
methodology, we explain how it can be used to model nonlinear L\'evy processes
in the sense of Neufeld and Nutz, and we introduce the new class of stochastic
partial differential equations under parameter uncertainty. Moreover, we study
properties of the nonlinear expectations. We prove the dynamic programming
principle, i.e., the tower property, and we establish conditions for the
(strong) $\textit{USC}_b$-Feller property and a strong Markov selection
principle.
