Once again this engineering brain is lost due to a lot of formal mathematics.
Consider the recursion $x(t+1) = A(t)x(t) + M(t)v(t)$
where $ x(t_0)=x_0$, $T = \{t_0,t_0 + 1,\ldots\}\subset\mathbb{Z}_+$, $x_0:\Omega \times \mathbb{R}^n$, $x_0 \in G(0,Q_0)$
EDIT: (meaning $x_0$ is a Gaussian random variable with mean $0$ and variance $Q_0$), and I assume that it should be $x_0:\Omega\to\mathbb{R}^n$, but there's an error in the exercise I guess,
$v:\Omega \times \mathbb{R}^{m_v}$, is a Gaussian white noise process with $v(t) \in G(0,Q_v)$, $\forall t\in T$,
$\mathcal{F}^{x_0}$, $\mathcal{F}^{v}_{\infty}$ are independent $\sigma$-algebras.
$A: T\to \mathbb{R}^{n\times n}$, $M:T\to\mathbb{R}^{n \times m_v}$, $Q_v \in \mathbb{R}^{m_v \times m_v}$,
$Q_v = Q^T_v > 0$,
$Q_x(t) = \mathbf{E} \left[ \left(x(t) - \mathbf{E}\left[x(t)\right] \right) \left( x(t) - \mathbf{E}\left[x(t)\right] \right)^T \right] > 0 $, $\forall t \in T$, assumed.
Derive a formula for $\mathbf{E}\left[x(t+2)|\mathcal{F}^{x(t)}\right]$,$\forall t\in \mathbb{Z}_+$ in terms of $x(t)$ and deterministic functions. Hint: use induction. This exercise needs concepts of Markov processes and of Gaussian processes.
Currently I've rewritten it in the following way: $$x(t+2) = A(t+1)\left(A(t)x(t) + M(t)v(t)\right) + M(t+1)v(t+1)$$
and I have tried different approaches, but somehow haven't figured out a way that seems to get near to the solution.