**Mathematica demonstration**: a simple application of rough paths to Fractional Orstein-Uhlenbeck process.

(click on the image to change the parameters interactively on Wolfram’s website.)

The first two moments (mean and variance) of an Ornsteinâ€“Uhlenbeck (OU) process are approximated with stochastic expansions (linear combinations of iterated integrals of the paths). The first three parameters are the usual parameters for an OU process: a high mean reversion makes the convergence to the mean faster and a high volatility increases the variance. The Hurst index controls the roughness of the fractional Brownian motion: the higher the value, the smoother the path. At Hurst index 0.5, this reduces to the usual Brownian motion.

This is a simple application of the theory of rough paths. The same approach can be used for a large class of drivers, not just fractional Brownian motion, and is applicable to more complicated models. Once a stochastic expansion is derived, it can be used for local simulation, moment approximations, or parameter estimation.