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In this work, we develop a multifactor approximation for $d$-dimensionalStochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels ofcompletely monotone type that may be singular. First, we prove an$L^2$-estimation between two SVEs with different kernels, which provides aquantification of the error between the SVE and any multifactor StochasticDifferential Equation (SDE) approximation. For the particular rough kernel casewith Hurst parameter lying in $(0,1/2)$, we propose various approximatingmultifactor kernels, state their rates of convergence and illustrate theirefficiency for the rough Bergomi model. Second, we study a Euler discretizationof the multifactor SDE and establish a convergence result towards the SVE thatis uniform with respect to the approximating multifactor kernels. Theseobtained results lead us to build a new multifactor Euler scheme that reducessignificantly the computational cost in an asymptotic way compared to the Eulerscheme for SVEs. Finally, we show that our multifactor Euler scheme outperformsthe Euler scheme for SVEs for option pricing in the rough Heston model.

Approximation of Stochastic Volterra Equations with kernels of completely monotone type