Weak error approximation for rough and Gaussian mean-reverting stochastic volatility models

For a class of stochastic models with Gaussian and rough mean-reverting volatility that embeds the genuine rough Stein-Stein model, we study the weak approximation rate when using a Euler type scheme with integrated kernels. Our first result is a weak convergence rate for the discretised rough Ornstein-Uhlenbeck process, that is essentially in $\min(3\alpha-1,1)$, where $\frac{t^{\alpha-1}}{\Gamma(\alpha)}$ is the fractional convolution kernel with $\alpha \in (1/2,1)$. Then, our main result is to obtain the same convergence rate for the corresponding stochastic rough volatility model with polynomial test functions.