Related papers: A theory of generalised coordinates for stochastic…

Modeling non-Markovian time series is a recent topic of research in many fields such as climate modeling, biophysics, molecular dynamics, or finance. The generalized Langevin equation (GLE), given naturally by the Mori-Zwanzig projection…

Neural Stochastic Differential Equations (NSDEs) model the drift and diffusion functions of a stochastic process as neural networks. While NSDEs are known to make accurate predictions, their uncertainty quantification properties have been…

We develop a class of non-Gaussian translation processes that extend classical stochastic differential equations (SDEs) by prescribing arbitrary absolutely continuous marginal distributions. Our approach uses a copula-based transformation…

In areas such as finance, engineering, and science, we often face situations that change quickly and unpredictably. These situations are tough to handle and require special tools and methods capable of understanding and predicting what…

This thesis is mainly concerned with state-space approaches for solving deep (temporal) Gaussian process (DGP) regression problems. More specifically, we represent DGPs as hierarchically composed systems of stochastic differential equations…

This work proposes a general framework for capturing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic…

We investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors. The nodes carry generalized charge that may be randomly transferred from one node to another. Our goal is to find the time…

Stochastic differential equations (SDEs) are of utmost importance in various scientific and industrial areas. They are the natural description of dynamical processes whose precise equations of motion are either not known or too expensive to…

In this article we study (possibly degenerate) stochastic differential equations (SDE) with irregular (or discontiuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere…

This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane-mediated receptor dynamics on cell surfaces, we…

A general formalism is developed to construct a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are therefore internal to the system and not externally specified. For…

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…

The Stochastic Partial Differential Equation (SPDE) approach, now commonly used in spatial statistics to construct Gaussian random fields, is revisited from a mechanistic perspective based on the movement of microscopic particles, thereby…

In the vanishing learning rate regime, stochastic gradient descent (SGD) is now relatively well understood. In this work, we propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime. The focus…

In this work, we study the numerical approximation of local fluctuations of certain classes of parabolic stochastic partial differential equations (SPDEs). Our focus is on effects for small spatially-correlated noise on a time scale before…

Correlation and smoothness are terms used to describe a wide variety of random quantities. In time, space, and many other domains, they both imply the same idea: quantities that occur closer together are more similar than those further…

I propose a novel framework that integrates stochastic differential equations (SDEs) with deep generative models to improve uncertainty quantification in machine learning applications involving structured and temporal data. This approach,…

Inspired by the stochastic particle method, this paper establishes an easily implementable explicit numerical method for McKean-Vlasov stochastic differential equations (MV-SDEs) with superlinear growth coefficients. The paper establishes…

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general c\`adl\`ag semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries…

Discovering the underlying relationships among variables from temporal observations has been a longstanding challenge in numerous scientific disciplines, including biology, finance, and climate science. The dynamics of such systems are…