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.

中文翻译：

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双因素模型中的有效波动率估计

我们统计分析了具有随机波动率的多元 HJM 扩散模型。第一个因子的波动率过程完全未指定，而第二个因子的波动率是未知过程和到期时间的指数函数的乘积。这个指数项包括一些实际参数，用于衡量随着时间的推移，第二个因素的增长速度。从历史数据中，我们在构建一个估计器的意义上有效地估计了成熟时间参数，该估计器在半参数设置中实现了最佳信息界限。我们还以非参数方式识别波动率过程的路径并实现极小极大界限。我们解决了当过程的维度大于二时发生的退化问题，并在漂移过程的适当规律性假设下给出特别的最优极限定理。我们始终如一地分析我们的估算器在电力市场远期合约价格的模拟和真实数据集上的数值行为。