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：我们研究随机波动率模型中波动率粗糙度的估计问题，该模型由带漂移的分数布朗运动的非线性函数生成。为此，我们提出一种新的估计量，基于连续轨迹反导数的离散观测来测量其所谓的粗糙指数。我们给出了底层轨迹满足的条件，使得该估计量在严格轨道意义上收敛。随后，我们验证了这些条件对分数布朗运动（带漂移）的几乎每条样本路径均成立。因此，我们在广泛粗糙波动率模型类中得到了强相合性定理。数值模拟表明，在对估计量进行尺度不变修正后，我们的估计方法表现良好。