  We consider rough stochastic volatility models where the variance process
satisfies a stochastic Volterra equation with the fractional kernel, as in the
rough Bergomi and the rough Heston model. In particular, the variance process
is therefore not a Markov process or semimartingale, and has quite low
H\"older-regularity. In practice, simulating such rough processes thus often
results in high computational cost. To remedy this, we study approximations of
stochastic Volterra equations using an $N$-dimensional diffusion process
defined as solution to a system of ordinary stochastic differential equation.
If the coefficients of the stochastic Volterra equation are Lipschitz
continuous, we show that these approximations converge strongly with
superpolynomial rate in $N$. Finally, we apply this approximation to compute
the implied volatility smile of a European call option under the rough Bergomi
and the rough Heston model.
