Expected Solution of a Stochastic Differential Equation as a Conditional Expectation (this is a tough one). On all you geniusses out there: this is a tough one.
Preliminaries and Rigorous Technical Framework


*

*Let $T \in (0, \infty)$ be fixed.

*Let $d \in \mathbb{N}_{\geq 1}$ be fixed.

*Let $$(\Omega, \mathcal{G}, (\mathcal{G}_t)_{t \in [0,T]},
   \mathbb{P})$$ be a complete probability space with a complete,
right-continuous filtration $(\mathcal{G}_t)_{t \in [0,T]}$.

*Let $$B : [0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad
   (t,\omega) \mapsto B_t(\omega)$$ be a standard $d$-dimensional
$(\mathcal{G}_t)_{t \in [0,T]}$-adapted Brownian motion on
$\mathbb{R}^d$ such that, for every pair $(t,s) \in \mathbb{R}^2$
with $0 \leq t < s$, the random variable $B_s-B_t$ is independent of
$\mathcal{G}_t$.

*Let  \begin{align}  &\sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d
   \times d}, \\  &\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d},
    \end{align}  be affine linear transformations, i.e. let there be matrices
$(A^{(\sigma)}_1,...,A^{(\sigma)}_d, \bar{A}^{(\sigma)}):= \theta_{\sigma} \in
   (\mathbb{R}^{d \times d})^{d+1}$ such that, for all $x \in \mathbb{R}^d$,
\begin{equation}  \sigma(x) = ( A^{(\sigma)}_1 x \mid ... \mid
   A^{(\sigma)}_d x) + \bar{A}^{(\sigma)},  \end{equation}  where
$A^{(\sigma)}_i x$ describes the $i$-th column of the matrix
$\sigma(x) \in \mathbb{R}^{d \times d}$, and let there be a matrix-vector pair $(A^{(\mu)}, \bar{a}^{(\mu)}) := \theta_{\mu}
   \in \mathbb{R}^{d \times d} \times \mathbb{R}^d$ 
such that, for all $x \in \mathbb{R}^d$,   \begin{equation}  \mu (x) =
   A^{(\mu)}x + \bar{a}^{(\mu)}. \end{equation} 

*Let
    \begin{equation}  \varphi : \mathbb{R}^d \rightarrow \mathbb{R}
       \end{equation}  be a fixed, continuous and at most polynomially growing function, i.e.    let $\varphi$ be continuous and let there be a constant $C
   \in [1,    \infty)$ such that, for all $x \in \mathbb{R}^d$ it holds
that
    \begin{equation}  \lVert \varphi(x) \rVert \leq C (1+\lVert x \rVert    )^C.  \end{equation}

*Let $x_0 \in \mathbb{R}^d$ be fixed.
Question
Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise:
\begin{equation}
 S_t = x_0 + \int_{0}^{t} \mu(S_t) ds + \int_{0}^{t} \sigma (S_t) dB_s.
\end{equation}
Under our assumptions, it is the case that an (up to indistinguishability) unique solution process 
$$ S^{(x_0, \theta_{\sigma}, \theta_{\mu})} :[0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t, \omega) \mapsto S_t(\omega),$$
for this equation exists (to see this, consider for example Theorem 8.3. in Brownian Motion, Martingales and Stochastic Calculus from Le Gall).
I am interested in the expectation of $S^{(x_0, \theta_{\sigma}, \theta_{\mu})}$ at time $T$ when passed through the function $\varphi$:
$$ \mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)].$$
More specifically, I want to express $\mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)]$ in the following way as a conditional expectation:
$$ \mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)] =
\mathbb{E}[\varphi(S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}_T) \mid (X_0, \Theta_{\sigma}, \Theta_{\mu}) = (x_0, \theta_{\sigma}, \theta_{\mu})]. $$
Here 
$$ X_0 : \Omega \rightarrow \mathbb{R}^d, $$
$$ \Theta_{\mu} : \Omega \rightarrow \mathbb{R}^{d \times d} \times \mathbb{R}^d,$$
$$ \Theta_{\sigma} : \Omega \rightarrow (\mathbb{R}^{d \times d})^{d+1},$$
 are $\mathcal{G}_0$-measurable random variables, which define the initial value $x_0$ of the process at $t=0$ as well as the entries of the affine-linear coefficient functions $\mu$ and $\sigma$. Moreover, $\Sigma$ is a random function. 
The random variable
$$ S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}_T : \Omega \rightarrow \mathbb{R}^d$$
is implicitly defined by the procedure of first "drawing" the random variables $(X_0, \Theta_{\sigma}, \Theta_{\mu})$ at time $t = 0$ in order to obtain fixed values 
$$ (X_0, \Theta_{\sigma}, \Theta_{\mu}) = (\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu}) $$ and then "afterwards" set
$$  S^{X_0, \Theta_{\sigma}, \Theta_{\mu})}_T :=  S^{(\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})}_T, $$
 where
$$ S^{(\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})} :[0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t, \omega) \mapsto S^{(\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})}_t(\omega) $$
is the (up to indistinguishability) unique solution process of the stochastic differential equation.
\begin{equation}
 S_t = \tilde{x}_0 + \int_{0}^{t} \tilde{\mu}(S_t) ds + \int_{0}^{t} \tilde{\sigma} (S_t) dB_s.
\end{equation}
Here, $\tilde{\sigma}$ and $\tilde{\mu}$ are the affine-linear maps associated with the parameter values $\tilde{\theta}_{\sigma}$ and $\tilde{\theta}_{\mu}$ as described above.
Now, my questions:


*

*I know that there are technical problems with the way I "defined " the random variable $S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}$, although I hope the idea is clear. How can I make the definition of $S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}$ rigorous in the above framework? 

*After having obtained a rigorous definition of $S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}$, how can I then show, that 
$$ \mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)] =
\mathbb{E}[\varphi(S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}_T) \mid (X_0, \Theta_{\sigma}, \Theta_{\mu}) = (x_0, \theta_{\sigma}, \theta_{\mu})] ?$$
If further regularity assumptions (for example on the random variables $X_0, \Theta_{\sigma}, \Theta_{\mu}$) are necessary in order to answer the above questions in a satisfactory way, then these can be made without second thoughts.
These questions are at the core of my current research. I am stuck and I would be extremely grateful for any advice!
 A: The Picard iteration method used in the reference you cite (look also at Theorem 8.5) shows that $(x_0,\theta_\sigma,\theta_\mu,\omega)\mapsto S_T^{(x_0,\theta_\sigma,\theta_\mu)}(\omega)$ is a jointly measurable function of $(x_0,\theta_\sigma,\theta_\mu,\omega)$, even continuous in the first three variables. The composite function $\omega \mapsto \varphi( S_T^{(X_0(\omega),\Theta_\sigma(\omega),\Theta_\mu(\omega))}(\omega))$ is therefore a $\mathcal G_T$-measurable random variable. (I'm assuming that $\varphi$ is bounded and Borel measurable.) Moreover, because $\mathcal G_0$ is independent of the increments of the driving Brownian motion $B$, the random variable $S_T^{(x_0,\theta_\sigma,\theta_\mu)}$ is independent of $\mathcal G_0$ for each fixed choice of $(x_0,\theta_\sigma,\theta_\mu)$. 
The identity you request is a special case of a more general (but easier to state when shorn of the SDE notation) fact.  Suppose $(z,\omega)\mapsto F(z,\omega)$ is bounded and jointly measurable (on $\Bbb R^n\times \Omega$, say) and that $Z: \Omega\to \Bbb R^n$ is a random variable independent of $\omega\mapsto F(z,\omega)$, for each fixed $z$. The composite function $G:\omega\mapsto F(Z(\omega),\omega)$ is then a random variable, and
$$
E[G\,|\,Z=z](\omega) = E[F(z,\cdot)],\qquad\qquad(\dagger)
$$
almost surely. That is, if you define $h(z):= E[F(z,\cdot)]$ then $h$ is Borel measurable (Fubini) and $\omega\mapsto h(Z(\omega))$ is a version of the conditional expectation on the left side of ($\dagger$). Apply this to $Z(\omega)=(X_0(\omega),\Theta_\sigma(\omega),\Theta_\mu(\omega)$ and $F(z,\omega)=F((x_0,\theta_\sigma,\theta_\mu),\omega) = \varphi( S_T^{(x_0,\theta_\sigma,\theta_\mu)}(\omega))$. The proof of ($\dagger$) is a matter of chasing the definition of conditional expectation, and Fubini's theorem.
