We fully characterize discrete-time dynamic convex expectations $(\mathcal{E}_t)$ with domain and range the upper semianalytic functions - in particular we work without a reference measure and do not assume essential suprema to exist. It is shown that $\mathcal{E}_t$ is pointwise continuous from below and continuous from above on the continuous functions if and only if a dual representation of $\mathcal{E}_t$ in terms of conditional expectations minus the convex conjugate of $\mathcal{E}_t$ holds true, where the conjugate is lower semianalytic with pointwise weakly compact level sets. Moreover, we provide a dual characterization of the dynamic property, i.e. we show that $\mathcal{E}_t=\mathcal{E}_t\circ\mathcal{E}_{t+1}$ if and only if the convex conjugate of $\mathcal{E}_t$ has an additive form. We also consider dynamic convex expectations defined on the set of discrete-time stochastic processes.
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