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These lectures notes aim at introducing L\'{e}vy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of L\'{e}vy processes. We analyze a `toy' example of a…
It is shown that the stochastic model of Fenyes and Nelson can be generalized in such a way that the diffusion constant of the Markov theory becomes a free parameter. This extra freedom allows one to identify quantum mechanics with a class…
Reaction-diffusion equations deliver a versatile tool for the description of reactions in inhomogeneous systems under the assumption that the characteristic reaction scales and the scales of the inhomogeneities in the reactant…
The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the L\'evy stable distributions, and more…
We characterize the Markovian and affine structure of the Volterra Heston model in terms of an infinite-dimensional adjusted forward process and specify its state space. More precisely, we show that it satisfies a stochastic partial…
In this paper, we study the convolution structure in the special affine Fourier transform domain to combine the advantages of the well known special affine Fourier and Stockwell transforms into a novel integral transform coined as special…
Linear fractional Galton-Watson branching processes in i.i.d.~random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals.…
Diffusion Models are probabilistic models that create realistic samples by simulating the diffusion process, gradually adding and removing noise from data. These models have gained popularity in domains such as image processing, speech…
Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term…
We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution…
We study gradient drift-diffusion processes on a probability simplex set with finite state Wasserstein metrics, namely finite state Wasserstein common noises. A fact is that the Kolmogorov transition equation of finite reversible Markov…
Point processes model the distribution of random point sets in mathematical spaces, such as spatial and temporal domains, with applications in fields like seismology, neuroscience, and economics. Existing statistical and machine learning…
For a general class of diffusion processes with multiplicative noise, describing a variety of physical as well as financial phenomena, mostly typical of complex systems, we obtain the analytical solution for the moments at all times. We…
Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We…
A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to control pathwisely the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp…
We consider a hidden Markov model, where the signal process, given by a diffusion, is only indirectly observed through some noisy measurements. The article develops a variational method for approximating the hidden states of the signal…
We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as…
In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point…
We establish circumstances under which the dispersion of passive contaminants in a forced, deterministic or random, flow can be consistently interpreted as a Markovian diffusion process. In case of conservative forcing the repulsive case…
Light scattering from resonantly or nearly resonantly excited systems, known as resonance fluorescence, has been gaining importance as a versatile tool for investigating quantum states of matter and readout of quantum information, recently…