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We compare the rate of convergence to the time average of a function over an integrable Hamiltonian flow with the one obtained by a stochastic perturbation of the same flow. Precisely, we provide detailed estimates in different Fourier…
We consider the stochastic continuity equation driven by Brownian motion. We use the techniques of the Malliavin calculus to show that the law of the solution has a density with respect to the Lebesgue measure. We also prove that the…
In this article we study a class of stochastic functional differential equations driven by L\'{e}vy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals.…
We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy…
The strong convergence of numerical methods for stochastic differential equations (SDEs) for $t\in[0,\infty)$ is proved. The result is applicable to any one-step numerical methods with Markov property that have the finite time strong…
We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution…
We introduce an infinite time horizon Brownian bridge which is determined by a stochastic Langevin equation with time dependent drift coefficient. We show that this process goes to zero almost surely when the time goes to infinity and study…
In this paper, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second order time-stepping for the numerical solution of the "good" Boussinesq equation.…
We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular,…
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method…
This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial…
The discrete-time approximation for nonlinear filtering problems is related to both of strong and weak approximations of stochastic differential equations. In this paper, we propose a new method of proof for the convergence of approximate…
This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the…
In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a H\"older continuous function with parameter greater that 1/2 via the Young integration setting. Then some…
Let $\mu$ = ($\mu$t)t$\in$R be a 1-parameter family of probability measures on R. In [11] we introduced its ``Markov-quantile''process: a process X= (Xt)t$\in$R that resembles as much as possible the quantile process attached to $\mu$,…
We examine the so-called micropolar equations in three dimensional cylindrical domains under Navier boundary conditions. These equations form a generalization of the ordinary incompressible Navier-Stokes model, taking the structure of the…
This work focuses on topics related to Hamiltonian stochastic differential equations with L\'{e}vy noise. We first show that the phase flow of the stochastic system preserves symplectic structure, and propose a stochastic version of…
We obtain the first probabilistic proof of continuous differentiability of time-dependent optimal boundaries in optimal stopping problems. The underlying stochastic dynamics is a one-dimensional, time-inhomogeneous diffusion. The gain…
We study the Rayleigh-Stokes problem for a generalized second-grade fluid which involves a Riemann-Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete…
We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of…