We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form ${\left|x\right|}^{\alpha}$, $\alpha \in [1/2,1)$. In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

In this paper we explicit the derivative of the flows of one-dimensional reflected diffusion processes. We then get stochastic representations for derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions.

We consider one-dimensional stochastic differential equations
in the particular case of diffusion coefficient functions of the form
|, ∈ [1/2,1). In that case, we study the rate of convergence of a
symmetrized version of the Euler scheme. This symmetrized version is
easy to simulate on a computer.
We prove its strong convergence and obtain the same rate of
convergence as when the coefficients are Lipschitz.

Motivated by the development of efficient Monte Carlo methods
for PDE models in molecular dynamics,
we establish a new probabilistic interpretation of a family of divergence form
operators with discontinuous coefficients at the interface
of two open subsets of ${\mathbb{R}}^{d}$. This family of operators includes the case of the
linearized Poisson-Boltzmann equation used to
compute the electrostatic free energy of a molecule.
More precisely, we explicitly construct a Markov process whose
infinitesimal generator...

This work aims at introducing modelling, theoretical and numerical studies related to a new downscaling technique applied to computational fluid dynamics.
Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor.
The local model, compatible with the Navier-Stokes equations, is used
for the small scale computation (downscaling) of the considered
fluid. It is
inspired by Pope's works on turbulence, and consists in...

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