  This paper introduces a novel It\^{o} diffusion process for both factor and
idiosyncratic volatilities whose eigenvalues follow the vector auto-regressive
(VAR) model. We call it the factor and idiosyncratic VAR-It\^{o} (FIVAR-It\^o)
model. The FIVAR-It\^o model considers dynamics of the factor and idiosyncratic
volatilities and involve many parameters. In addition, the empirical studies
have shown that the financial returns often exhibit heavy tails. To address
these two issues simultaneously, we propose a penalized optimization procedure
with a truncation scheme for a parameter estimation. We apply the proposed
parameter estimation procedure to predicting large volatility matrices and
investigate its asymptotic properties. Using high-frequency trading data, the
proposed method is applied to large volatility matrix prediction and minimum
variance portfolio allocation.
