Definition (Static Quadratic Team)

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:
3. $R_{ij}(\xi)$ is a matrix-valued Random Variable,
4. $r_{i}(\xi)$ is a Random Vector, and
5. $c(\xi)$ is a random variable,

all generated by measurable mappings on the random state of nature $\xi$.

1. $L(\xi;\mathbf{u})$ 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.)
2. $R(\xi)$ is uniformly bounded above i.e. $\exists \beta>0$ s.t. $\beta I-R(\xi)$ is positive definite a.s.
3. $\mathbb{Y}^{i}\equiv \mathbb{R}^{m_{i}},i\in\mathcal{N}$, (i.e. the measurement spaces are unconstrained Euclidean spaces.)
4. $y^{i}=\eta^{i}(\xi),i\in\mathcal{N}$ for some appropriate Borel measurable functions $\eta^{i},i\in\mathcal{N}$.
5. $\Gamma^{i}$ is the (Hilbert) space of all Borel measurable mappings of $\gamma^{i}:\mathbb{R}^{r_{i}}\to \mathbb{R}^{m_{i}}$, which are in $L^{2}(\Omega,\mathcal{F},P)$.
6. $r_{i}(\xi)\in L^{2}(\Omega,\mathcal{F},P)$ $c(\xi)\in L^{1}(\Omega,\mathcal{F},P)$ 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.

Definition (Standard quadratic team)

Let $\{ J;\Gamma^{i},i\in\mathcal{N} \}$ be a LQG Teams. We call this a standard quadratic team if furthermore the matrix $R$ is constant for all $\xi$ (i.e. it is deterministic).

Definition (Quadratic Gaussian team)

Let $\{ J;\Gamma^{i},i\in\mathcal{N} \}$ be a . If $\xi$ is a Gaussian random vector and $r_{i}(\xi)=Q_{i}\xi$, $\eta^{i}(\xi)=H^{i}\xi$, $i\in\mathcal{N}$, for some deterministic matrices $Q_{i},H^{i},i\in\mathcal{N}$, the decision problem is a quadratic-Gaussian team.