Convexity of a random-walk Snell value function ($W_t(x)=\max \{f(x),\mathbb{E}[W_{t+1}(x+\xi )]\}$)

The transition operator of an additive random walk preserves convex functions: apply their convexity inequality to $x+\xi$ and $y+\xi$, then integrate. The pointwise maximum of convex functions preserves convexity too, so induction proves convexity of the Snell envelope Bellman functions. Expectations are allowed to be extended-valued until finiteness is justified. Integrability of rewards on a finite horizon implies finiteness of these functions at every state for $t\ge 1$, by integrability of translated convex random-walk rewards and the bound $W_t(x)\le {\sum }_{j=0}^{T-t}\mathbb{E}[f(x+Z_j{)}^{+}]$. At time zero only $X_0=0$ is observed; if $W_0$ is infinite elsewhere, its finite value at zero has a constant convex extension representing the same Snell envelope. Full statewise Bellman recursion at time zero requires an additional translated-integrability hypothesis.