Definition

Given a probability space $(\Omega,\mathcal{F},P)$, and an associated Random Vector $\xi$, let $\{ J;\Gamma^{i},i\in\mathcal{N} \}$ be a static stochastic team problem with the following specifications: 1. $\mathbb{U}^{i}\equiv\mathbb{R}^{m_{i}},i\in\mathcal{N}$ i.e. the action spaces are unconstrained Euclidean spaces. 2. The loss function is a quadratic function of $\mathbf{u}$ for every $\xi$: $$L(\xi;\mathbf{u}):=\sum_{i,j\in\mathcal{N}}u^{i'}R_{ij}(\xi)u^{j}+2\sum_{i\in\mathcal{N}}u^{i'}r_{i}(\xi)+c(\xi)$$where: 1. $R_{ij}(\xi)$ is a matrix-valued Random Variable, 2. $r_{i}(\xi)$ is a Random Vector, and 3. $c(\xi)$ is a random variable, all generated by measurable mappings on the random state of nature $\xi$. 3. $L(\xi;\mathbf{u})$ is strictly (and uniformly) convex in $\mathbf{u}$ a.s. i.e. $\exists\alpha>0$ s.t. $R(\xi)$ defined as a matrix composed of $N$ blocks, with the $ij$’th block given by $R_{ij}(\xi)$, the matrix $R(\xi)-\alpha I$ is positive definite a.s. 4. $R(\xi)$ is uniformly bounded above i.e. $\exists \beta>0$ s.t. $\beta I-R(\xi)$ is positive definite a.s. 5. $\mathbb{Y}^{i}\equiv \mathbb{R}^{m_{i}},i\in\mathcal{N}$, i.e. the measurement spaces are unconstrained Euclidean spaces. 6. $y^{i}=\eta^{i}(\xi),i\in\mathcal{N}$ for some appropriate Borel measurable functions $\eta^{i},i\in\mathcal{N}$. 7. $\Gamma^{i}$ is the (Hilbert) space of all Borel measurable mappings of $\gamma^{i}:\mathbb{R}^{r_{i}}\to \mathbb{R}^{m_{i}}$, which are in $L^{2}(\Omega,\mathcal{F},P)$. 8. $r_{i}(\xi)\in L^{2}(\Omega,\mathcal{F},P)$ $c(\xi)\in L^{1}(\Omega,\mathcal{F},P)$ i.e. $$E_{\xi}\left[ r_{i}'(\xi)r_{i}(\xi) \right]<\infty,i\in\mathcal{N}\quad E_{\xi}\left[ c(\xi) \right]<\infty$$ We call a static stochastic team quadratic if it satisfies the above conditions.