# Simulating volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes

I am trying to simulate the volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes. In fact, giving $$\mathbf{X}$$ a $$n$$-dimensional vector valued OU process with $$\begin{equation} \mathrm{d}X_t^i = \alpha X_t^i \mathrm{d}t + \beta \mathrm{d}W_t^i, \end{equation}$$ where $$\mathbf{W}$$ is a $$n$$-dimensional vector of independent Brownian motions.

Then, we know that the process $$\begin{equation} Y_t = \sum_{i = 1}^n \left( X_t^i \right)^2. \end{equation}$$

is a CIR process, such that

$$\begin{eqnarray} \mathrm{d}Y_t & = & \left( 2 \alpha Y_t + n \beta^2 \right) \mathrm{d}t + 2 \beta \sqrt{Y_t} \mathrm{d}\widetilde{W}_t\\ & = & \kappa \left( \theta - Y_t \right) \mathrm{d}t + \xi \sqrt{Y_t} \mathrm{d} \widetilde{W}_t, \end{eqnarray}$$

where $$\kappa = -2 \alpha$$, $$\theta = -n \beta^2 / 2 \alpha$$, $$\xi = 2 \beta$$ and $$\begin{equation} \widetilde{W}_t = \int_0^t \frac{1}{\sqrt{Y_u}} \sum_{i = 1}^n X_u^i \mathrm{d}W_u^i. \end{equation}$$

For n=1, the simulation is clear since basically in that case $$\widetilde{W}_t=W_t$$. However, it is not clear to me how to do the simulation for $$n>1$$ ($$n$$ integer) since, for the Heston model, I also need to construct the noise driving the asset price which is needed to be correlated to $$\widetilde{W}_t$$?! I know that I can just simulate $$\widetilde{W}_t$$ from its expression above but I thought there maybe a more efficient way than that! Any guidance for this issue? Many thanks!