RoughBench is a benchmark dataset for machine learning on rough differential equations.

The Ornstein-Uhlenbeck (OU) process is a classical mean-reverting stochastic process, governed by the SDE: $$ dX_t = \theta (\mu - X_t) dt + \sigma dW_t, $$ where $\theta$ is the rate of mean reversion, $\mu$ is the long-term mean, $\sigma$ is the volatility, and $W_t$ is standard Brownian motion.

A batch of OU processes simulated with $\theta=0.5$, $\mu=0.0$, $\sigma=0.3$.

The rough OU process replaces standard Brownian motion $W_t$ with a fractional Brownian motion $B^H_t$ (with Hurst parameter $H < 0.5$), capturing rougher, more persistent path behavior: $$ dX_t = \theta (\mu - X_t) dt + \sigma dB^H_t. $$ Below is a simulation with $\theta=0.5$, $\mu=0.0$, $\sigma=0.3$, and Hurst parameter $H=0.7$.

Black-Scholes is a fundamental model for option pricing that assumes the price of the underlying follows a geometric Brownian motion.

The Bergomi model is a stochastic volatility model known for capturing volatility clustering. It extends classical models by including a stochastic process for the variance, incorporating memory effects commonly observed in financial markets.

The rBergomi (rough Bergomi) model is an extension of the Bergomi model, introducing roughness in volatility paths using fractional Brownian motion. The roughness parameter (Hurst exponent) allows a more realistic fit to observed market volatility, especially in short-term smile/skew regimes.