Probabilistic rough paths

The theory of regularity structures, first developed by Martin Hairer (2014 Fields medal) explains how many problems with highly oscillatory driving signals can have incremental solutions which are described by an abstract Taylor expansion. There is a path on the model space that represents the desired solution and a model (a path on the characters of the model space) that acts on the solution to describe the solution increments in terms of a jet expansion.

In particular, a rough path can be considered as a model and the solution of a rough differential equation (a controlled rough path) has increments that are uniquely described by the increments of the rough path and an appropriately smooth remainder term.

A probabilistic rough path is another example of a model for the regularity structure derived from a Lions Taylor expansion. The Lions derivative provides a calculus for functions on the Wasserstein space of measures that characterises infinitesimal perturbations of a measure by infinitesimal variations on a linear space of random variables.

The means that probabilistic rough paths are the correct object for considering rough differential equations where the coefficients are distribution dependent. Extra information about the distribution of the driving signal is necessary in order to provide a space on which the solution map is continuous.

The model space is indexed by Lions trees, a non-planar rooted tree with an additional hypergraphic structure that captures the probabilistic structure of iterated integrals of the driving signal. The study of Lions trees has lead to a rich collection of new algebraic and analytic structures including Coupled Hopf algebras and the McKean-Vlasov group of characters.

Large Deviations Principles

Large Deviations Principles (LDPs) concerns the asymptotic behaviour of a sequences of probability measures acting on some event. When the sequence of measures are Gaussian distributions with fixed mean and variance that converges to 0, it is now well understood that the probability of events that contain the mean converge to 1 and the probability of events that do not contain the mean converge to 0. Further, the probability of events that do not contain the mean can be upper and lower bounded by a Gaussian tail distribution.

In many statistical phenomena, a Gaussian tail distribution represents a natural choice for the probability of a highly unlikely event and establishing an LDP formalises this by providing a rate function that gives the explicit upper and lower bound.

My reserach has focused on Freidlin-Wentzell LDPs where the limit is the magnitude of the variance of a driving signal being taken to 0. Such results demonstrate that when the order of the driving signal is small, the system is a small normal perturbation away from the ordinary differential equation obtained when the driving signal is replace by the constant 0 path.

Rough volatility models

Rough volatility models are a class of stochastic processes that are used in mathematical finance to model asset prices. These models are advantageous because they reproduce observed skew of implied volatility and have grown in popularity since it was observed that they arise as the scaling limit of micro-structure models under very natural conditions.

However, despite many of their relatively simplistic description, such models are not semi-martingales and so cannot be considered using classical tools from stochastic analysis. The most simple example of a rough volatility model is the ito integral of a smooth function of the sample path of a Fractional Brownian motion. Hence, the volatility is rougher than the driving Brownian motion.

A vital contribution to the field came when it was observed that the regularity structure of such a stochastic process (very different from a classical rough path) was necessary to calculate and estimate a strong solution for rough volatility models. However, this left the question of the weak error for a numerical scheme wide open.

One might naively think that the weak error is easier to solve than the strong error, but historically this has not been the case because calculating the weak error involves arduous calculations of Gaussian random variables. In an upcoming pre-print, Thomas Wagerhofer Peter Friz and I were able to use techniques from Malliavin calculus inspired by [Gassiat 2022] to explicitly compute and upper bound these terms in order to calculate a weak error bound. Curiously, we were able to confirm that the weak error is of order 3H+0.5 for a wide class of volatility functions.