We consider the problem of estimating the roughness of the volatility in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

翻译：我们考虑在由带漂移的分数布朗运动的非线性函数生成的随机波动率模型中，估计波动率粗糙度的问题。为此，我们提出了一种新的估计量，该估计量基于连续轨迹的反导数的离散观测，测量所谓的轨迹粗糙度指数。我们给出了底层轨迹的条件，在这些条件下，我们的估计量以严格的路径意义收敛。随后，我们验证了这些条件对于几乎所有的带漂移的分数布朗运动样本路径都成立。由此，我们在一大类粗糙波动率模型的背景下获得了强一致性定理。数值模拟表明，在对我们的估计量进行尺度不变修正后，我们的估计方法表现良好。