Applications in biology, finance, and physics often require modelling uncertain events such as the birth and death of particles, infection and recovery from diseases within a population, uncertainty in large particle systems in statistical physics, and the future value of assets in financial modelling. These events are inherently random and are typically described by probabilistic models. It is often assumed that these random events are independent of each other, meaning that future states are fully determined by the current state of the system. This lack of memory in the system is characteristic of Markov processes, for which a powerful mathematical framework exists.

However, many modern applications do not conform to Markov processes as they explicitly require models that account for memory effects. Such behaviour is observed, for example, in the SIR model in epidemiology, fractional Langevin equations and anomalous diffusion in physics, rough volatility models and spot price dynamics in electricity markets in Finance, and multi-scale climate change models. Therefore, my current research focuses on stochastic processes that incorporate these memory effects through a newly introduced Volterra kernel.

Stochastic Volterra processes present significant challenges due to their non-Markovian nature, which precludes the use of powerful analytic tools from semigroup theory and Kolmogorov equations. Moreover, if the Volterra kernel is rough, the corresponding process is not a semimartingale, ruling out classical methods from stochastic analysis such as the Itô calculus and Girsanov transformation. These fundamental obstacles when compared to Markov processes and classical semimartingale theory necessitate the development of new analytic techniques and tools to effectively capture and analyse the memory and roughness effects of the system.

Ergodic properties and limit theorems address the question of when corresponding processes admit an equilibrium (invariant measure), the stationarity of the time series generated by such processes, and corresponding limit theorems (e.g. law-of-large numbers, central limit theorems, large deviations). While for Markov processes, an extensive theory was developed in the past decades, much less is known for the non-markovian counterparts of stochastic Volterra processes. Such properties are of fundamental importance for model justification (e.g. stochastic volatility is mean-reverting) but also form the cornerstone of statistical inference techniques. In my research, I focus on the following central objectives:

• What is the correct notion of equilibria for non-Markov processes and how can one characterize the latter in terms of the memory inherited from the Volterra kernel? 

• How to efficiently estimate the parameters of the model from historical data? Is it possible to infer the model from historical data in a non-parametric way?

To answer this question, it turns out that additional analytical knowledge on the regularity of the distributions is crucial. The latter is encoded in the laws of (conditional) finite-dimensional distributions, the regularity of corresponding local times, and support properties of the processes. Thus, my research activity addresses the following questions that are also of independent interest beyond applications to limit theorems and statistical inference:

• What can be said about the smoothness/roughness of finite-dimensional distributions or associated local times? What changes for models with degeneracy (e.g. noise vanishes at the boundary)?

• Do such processes depend continuously (smoothly) on the past?