#StochasticControl # Theorem For an $N-$agent static stochastic team problem satisfying the four hypotheses, there exists at least one team-optimal solution.

Corollary

We can relax $(c.1)$ into $(c.1')$ and the result still holds: 1. $(c.1')$ Let $\mathcal{N}_{h}$ and $\mathcal{N}_{s}$ be two complementary subsets of $\mathcal{N}$ (i.e. $\mathcal{N}_{h}\cup \mathcal{N}_{s}=\mathcal{N}$, and $\mathcal{N}_{h}\cap \mathcal{N}_{s}=\emptyset$) s.t. $S^{i}$ is compact $\forall i\in\mathcal{N}_{h}$ and $S^{j}\equiv \mathbb{U}^{j}$ $\forall j\in\mathcal{N}_{s}$. Assume that $\sum_{j\in \mathcal{N}_{s}}|u^{j}|\to \infty,\ L(\xi;u^{1},\dots,u^{N})\to \infty$ a.s., for every fixed $u^{i}\in S^{i},i\in \mathcal{N}_{h}$.