= Convexity of a random-walk Snell value function
{title2=$W_t(x)=\max\{f(x),\mathbb E[W_{t+1}(x+\xi)]\}$}
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 $t\geq1$, by <integrability of translated convex random-walk rewards> and the bound $W_t(x)\leq\sum_{j=0}^{T-t}\mathbb E[f(x+Z_j)^+]$. 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.
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