I have a stochastic process with additive noise on the unit sphere, parametrized by angles (i.e. hyper-spherical coordinates with $r=1$): $$ \theta_t = \psi(Z_t)dt + \sigma(Z_t,\theta_t)dW_t \tag{1} $$ where $Z_t$ is a different process and $\theta_t\in\mathbb{R}^{n-1}$. We can convert to Cartesian coordinates via $ C_t = C(\theta_t) $, where $\mathbb{R}^n$. The nice fact is that using $\theta_t$ stays on the unit sphere, but in this case $\sigma$ has a singularity in spherical coordinates.
So I was wondering what happens if I do the following, as a numerical discretization (like a corrected Euler-Maruyama): $$ \tilde{C}_{t+1} = C_t + \eta(Z_t) \Delta t + \tilde{\sigma}(Z_t,C_t)\Delta W_t $$ $$ C_{t+1} = \frac{\tilde{C}_{t+1}}{||\tilde{C}_{t+1}||_2} $$ where $\tilde{\sigma}(Z_t,C_t)$ and $\eta(Z_t)$ are the Cartesian forms of $\sigma$ and $\psi$, and $\tilde{C}$ is the "initial" unnormalized version of $C_t$.
Questions: how much error will this incur as opposed to discretizing eq (1) directly? Will this converge to the same process for small enough $\Delta t$?