  We statistically analyse a multivariate HJM diffusion model with stochastic
volatility. The volatility process of the first factor is left totally
unspecified while the volatility of the second factor is the product of an
unknown process and an exponential function of time to maturity. This
exponential term includes some real parameter measuring the rate of increase of
the second factor as time goes to maturity. From historical data, we
efficiently estimate the time to maturity parameter in the sense of
constructing an estimator that achieves an optimal information bound in a
semiparametric setting. We also identify nonparametrically the paths of the
volatility processes and achieve minimax bounds. We address the problem of
degeneracy that occurs when the dimension of the process is greater than two,
and give in particular optimal limit theorems under suitable regularity
assumptions on the drift process. We consistently analyse the numerical
behaviour of our estimators on simulated and real datasets of prices of forward
contracts on electricity markets.
