Related papers: Martingale representations for diffusion processes…
In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of linear parabolic initial-and final boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary…
Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic…
Lions and Musiela (2007) give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert (2009) and Mijatovi\'c and Urusov (2012c)…
We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the…
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the…
In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDE), whose generators are continuous with linear growth. It generalizes some known representation theorems…
Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a…
In this paper we study the class of backward doubly stochastic differential equations (BDSDEs, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial…
The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale.…
The proposed model modifies option pricing formulas for the basic case of log-normal probability distribution providing correspondence to formulated criteria of efficiency and completeness. The model is self-calibrating by historic…
In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and…
We introduce a new class of reflected backward stochastic differential equations with two c\`adl\`ag barriers, which need not satisfy any separation conditions. For that reason, in general, the solutions are not semimartingales. We prove…
The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for…
We show that a one-dimensional regular continuous Markov process \(\X\) with scale function \(s\) is a Feller--Dynkin process precisely if the space transformed process \(s (X)\) is a martingale when stopped at the boundaries of its state…
We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special…
A model for diffusion on a cubic lattice with a random distribution of traps is developed. The traps are redistributed at certain time intervals. Such models are useful for describing systems showing dynamic disorder, such as ion-conducting…
Point processes often have a natural interpretation with respect to a continuous process. We propose a point process construction that describes arrival time observations in terms of the state of a latent diffusion process. In this…
In this paper we will give a categorical proof of the Radon-Nikodym theorem. We will do this by describing the trivial version of the result on finite probability spaces as a natural isomorphism. We then proceed to Kan extend this…
We establish the existence of solutions to a class of non-linear stochastic differential equation of reaction-diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained…
Diffusion models have demonstrated remarkable capabilities in synthesizing realistic images, spurring interest in using their representations for various downstream tasks. To better understand the robustness of these representations, we…