  We study multidimensional stochastic volatility models in which the
volatility process is a positive continuous function of a continuous
multidimensional Volterra process that can be not self-similar. The main
results obtained in this paper are a generalization of the results due, in the
one-dimensional case, to Cellupica and Pacchiarotti [M. Cellupica and B.
Pacchiarotti (2021) Pathwise Asymptotics for Volterra Type Stochastic
Volatility Models. Journal of Theoretical Probability, 34(2):682--727]. We
state some (pathwise and finite-dimensional) large deviation principles for the
scaled log-price and as a consequence some (pathwise and finite-dimensional)
short-time large deviation principles.
