Since the introduction of rough volatility there have been numerous attempts at combining it with existing models in order to better approximate the volatility surface with a low number of parameters. The drawback of rough volatility is usually the time needed to compute a volatility surface. We compare three major rough volatility models and compare their ability to fit to the market volatility surface. We implement the rough Bergomi, rough Heston and lifted Heston models
and introduce a fourth model by taking a Nelson-Siegel parameterization for the instantaneous forward variance curve in the rough Bergomi model. We then minimize the volatility surfaces of the models to that of the market and compare these results through minimization time, fit and hedging possibilities. We find that for both Bergomi models, a single volatility surface is seven times faster to compute than for the lifted Heston model, which in turn is five times faster to compute than for the rough Heston model. Minimization times for the rough Heston models are comparable, however significantly higher than for the rough Bergomi models. We also find that the rough Bergomi model is under-parameterised, this is however fixed by the Nelson-Siegel parameterization, which has a minimization error in line with that of both Heston models. Finally, hedging specifically against a parameter rarely improves the outcome, although a hedge against the instantaneous forward variance curve is consistently good and can improve the error between only a delta hedge by 25-50\% throughout all models.

In this thesis a simple model of an active particle was taken, one including a drift according to its internal state. For this model a diffusion coefficient was determined as well as a rate function for the large deviations. This same model is also applied to electrons, although these particles are not active a diffusion coefficient was also found numerically. Finally there is a section on finding the ground state energy of drifting particles with a Hubbard-U term using diffusion Monte Carlo