Easy Mistake With the Log-Euler Discretization On Black-Scholes

In the Black-Scholes model with a term-structure of volatilities, the Log-Euler Monte-Carlo scheme is not necessarily exact. This happens if you have two assets (S_1) and (S_2), with two different time varying volatilities (\sigma_1(t), \sigma_2(t) ). The covariance from the Ito isometry from (t=t_0) to (t=t_1) reads $\int_{t_0}^{t_1} \sigma_1(s)\sigma_2(s) \rho ds,$ while a naive log-Euler discretization may use $\rho \bar\sigma_1(t_0) \bar\sigma_2(t_0) (t_1-t_0).$ In practice