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Stochastic processes are often used to model complex scientific problems infields ranging from biology and finance to engineering and physical science.This paper investigates rate-optimal estimation of the volatility matrix of ahigh-dimensional It\^{o} process observed with measurement errors at discretetime points. The minimax rate of convergence is established for estimatingsparse volatility matrices. By combining the multi-scale and thresholdapproaches we construct a volatility matrix estimator to achieve the optimalconvergence rate. The minimax lower bound is derived by considering a subclassof It\^{o} processes for which the minimax lower bound is obtained through anovel equivalent model of covariance matrix estimation for independent butnonidentically distributed observations and through a delicate construction ofthe least favorable parameters. In addition, a simulation study was conductedto test the finite sample performance of the optimal estimator, and thesimulation results were found to support the established asymptotic theory.

Optimal sparse volatility matrix estimation for high-dimensional It\^{o} processes with measurement errors