Related papers: Moderate and $L^p$ maximal inequalities for diffus…
The multivariate Ornstein-Uhlenbeck process is used in many branches of science and engineering to describe the regression of a system to its stationary mean. Here we present an $O(N)$ Bayesian method to estimate the drift and diffusion…
Lecture notes for a master-level mathematics course on martingales and stochastic calculus, held at the University of Orl\'eans, France. With corrected exercises. Contents: Discrete-time martingales, stopping times, convergence theorems.…
We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates…
Consider a discrete-time martingale, and let $V^2$ be its normalized quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian…
In this paper, we provide strong $L_2$-rates of approximation of the integral-type functionals of Markov processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its…
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be…
Two aspects of noncolliding diffusion processes have been extensively studied. One of them is the fact that they are realized as harmonic Doob transforms of absorbing particle systems in the Weyl chambers. Another aspect is integrability in…
In this paper we consider an Ornstein-Uhlenbeck (OU) process $(M(t))_{t\geqslant 0}$ whose parameters are determined by an external Markov process $(X(t))_{t\geqslant 0}$ on a finite state space $\{1,\ldots,d\}$; this process is usually…
We consider a reflected Ornstein-Uhlenbeck process $X$ driven by a fractional Brownian motion with Hurst parameter $H\in (0, \frac12) \cup (\frac12, 1)$. Our goal is to estimate an unknown drift parameter $\alpha\in (-\infty,\infty)$ on the…
In this paper, we investigate the optimal control problem for systems driven by mixed fractional Brownian motion (including a fractional Brownian motion with Hurst parameter $H>1/2$ and the standard Brownian motion). By using Malliavin…
We study Fourier multipliers which result from modulating jumps of L\'evy processes. Using the theory of martingale transforms we prove that these operators are bounded in $L^p(\Rd)$ for $1<p<\infty$ and we obtain the same explicit bound…
We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy…
The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron-Martin space. It…
Let $(x_k)_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras…
In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale…
This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using…
We consider stochastic differential equations in a Hilbert space, perturbed by the gradient of a convex potential. We investigate the problem of convergence of a sequence of such processes. We propose applications of this method to…
We construct a class of superprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching…
We investigate the sharp large deviation properties of the energy and the maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst index greater than one half.
Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained for maximal operators over a wide range of…