Related papers: Burkholder's submartingales from a stochastic calc…
We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the…
We discuss physical and mathematical aspects of the over-damped motion of a Brownian particle in fluctuating potentials. It is shown that such a system can be described quantitatively by fluctuating rates if the potential fluctuations are…
We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A noncommutative counterpart of a classical process continuous with probability one is introduced, and a…
Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in…
We investigate in this work the spectrum of singularities of super-Brownian motion with stable branching. The main purpose is to provide a uniform description of the latter in high dimension $d\geq\tfrac{2}{\gamma-1}$, presenting the…
We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some…
We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way…
A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with L\'evy process are investigated. We establish a comparison theorem which allows us to derive an…
The objective of this paper is to derive a representation of symmetric G-martingales as stochastic integrals with respect to the G-Brownian motion. For this end, we first study some extensions of stochastic calculus with respect to…
We provide a characterization of continuous semimartingales whose law is invariant with respect to predictable random rotations. In particular we prove that all such semimartingales are obtained by integrating a predictable process with…
We prove a new Burkholder-Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $(S(t,s))_{0\leq s\leq T}$ is a $C_0$-evolution family of contractions on a…
In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by $\alpha/2$-subordinated cylindrical Brownian motions with $\alpha\in(1,2)$. To prove the results, we truncate the…
Gradient optimization algorithms using epochs, that is those based on stochastic gradient descent without replacement (SGDo), are predominantly used to train machine learning models in practice. However, the mathematical theory of SGDo and…
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts…
Results on the existence, uniqueness and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are established. The goal is to develop a general multi-asset framework encompassing a wide spectrum of…
Let ${\psi}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $\psi\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space…
In this paper we show that solutions of stochastic partial differential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic…
The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod…
"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schr\"odinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually…
Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Levy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance.…