Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(t))dt+\gamma dW^{r}(t)$$
with constant scalars $S_{0}>0$, $T>0$, $r_{0}>0$, $\sigma>0$, $\theta>0$, $\gamma>0$, $\kappa>0$, and $t>0$, $dW^{S}(t)dW^{r}(t)=\rho dt$ with $t\in[0,T]$.
The exact solution to the stock price at time $T$ is as following
$$S(T)=S_{0}\exp(\int^{T}_{0}r(s)ds-\frac{1}{2}\sigma^{2}T+\sigma W^{S}(T))$$
By drawing $N$ times from $W(T)\sim\mathcal{N}(0,T)$ an approximation of the expected value can be made through a Monte Carlo simulation; however, the term $\int^{T}_{0}r(s)ds$ is stochastic, since the exact solution for $r(s)$ for the Vasicek model is as following
$$r(s)=r_{0}e^{-\kappa s}+\theta(1-e^{-\kappa s})+\gamma e^{-\kappa s}\int^{s}_{0}e^{\kappa\bar{s}}dW^{r}(\bar{s})$$
with $\bar{s}$ a dummy variable and $\int^{s}_{0}e^{\kappa\bar{s}}dW^{r}(\bar{s})\sim\mathcal{N}(0,\frac{1}{2\kappa}(e^{2\kappa s}-1))$ according to Ito's lemma, therefore
$$\int^{T}_{0}r(s)ds=\frac{r_{0}}{\kappa}(1-e^{\kappa T})+\theta T+\frac{\theta}{\kappa}(e^{\kappa T}-1)+\int^{T}_{0}\gamma e^{-\kappa s}\sqrt{\frac{1}{2\kappa}(e^{2\kappa s}-1)}Zds$$
with $Z\sim\mathcal{N}(0,1)$. Next, using integration by parts and substitution the integral $\int^{T}_{0}\gamma e^{-\kappa s}\sqrt{\frac{1}{2\kappa}(e^{2\kappa s}-1)}$ can be solved by firstly choosing $dU=\gamma e^{-\kappa s}$ and $V=\sqrt{\frac{1}{2\kappa}(e^{2\kappa s}-1)}$, which leads to
$$\int VdU=UV-\int UdV\implies\int^{T}_{0}\gamma e^{-\kappa s}\sqrt{\frac{1}{2\kappa}(e^{2\kappa s}-1)}ds=\frac{-\gamma e^{-\kappa T}}{\kappa}\sqrt{\frac{1}{2\kappa}(e^{2\kappa T}-1)}-\frac{\gamma}{2\kappa}\int^{T}_{0}\frac{1}{\sqrt{e^{\kappa s}-1}}ds$$
and secondly choosing $x=e^{\kappa s}-1$ with $ds=\frac{1}{\kappa x}dx$ leading to
$$\frac{-\gamma e^{-\kappa T}}{\kappa}\sqrt{\frac{1}{2\kappa}(e^{2\kappa T}-1)}-\frac{\gamma}{2\kappa}\int^{T}_{0}\frac{1}{\sqrt{e^{\kappa s}-1}}ds=\frac{-\gamma e^{-\kappa T}}{\kappa}\sqrt{\frac{1}{2\kappa}(e^{2\kappa T}-1)}+\frac{\gamma}{\kappa^{2}}(e^{\frac{-\kappa T}{2}}-1)$$
which is deterministic. Finally, a Monte Carlo simulation can be done by drawing $N$ times from $\int^{T}_{0}r(s)ds\sim\mathcal{N}(\mu=\frac{r_{0}-\theta}{\kappa}(1-e^{\kappa T})+\theta T,\sigma=\frac{-\gamma e^{-\kappa T}}{\kappa}\sqrt{\frac{1}{2\kappa}(e^{2\kappa T}-1)}+\frac{\gamma}{\kappa^{2}}(e^{\frac{-\kappa T}{2}}-1)$ to approximate the expectation of the exact solution of $S(T)$.
My questions are: i) is the above reasoning correct, where I write the integral with respect to $dW^{r}(t)$ as an integral with respect to $ds$ multiplied by a standard normal random variable, $Z$ and ii) since W^{r}(T) is now governed by $Z\sim\mathcal{N}(0,1)$ can I still compute $W^{S}(T)$ as following
$$W^{S}(T)=\sqrt{T}(\rho Z+\sqrt{1-\rho^{2}\bar{Z}})$$
with again $\bar{Z}\sim\mathcal{N}(0,1)$? (This is an uncoventional notation, but convenient in my opinion.)
