In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of ${ℂ}^n$, fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part $S$ which ensures that if such a perturbation of $S$ is formally conjugate to $S$ then it is also holomorphically conjugate to it. We study the normal form problem relatively to $S$. We give a condition on $S$ that ensures that there...

The properties of statistical solutions for some general differential equations in Banach spaces are investigated.

We define approximation schemes for generalized backward stochastic differential systems, considered in the Markovian framework. More precisely, we propose a mixed approximation scheme for the following backward stochastic variational inequality: $$d{Y}_t+F(t,X_t,Y_t,Z_t)dt\in \partial \phi \left(Y_t\right)dt+Z_{t}d{W}_t,$$ where ∂φ is the subdifferential operator of a convex lower semicontinuous function φ and (X t)t∈[0;T] is the unique solution of a forward stochastic differential equation. We use an Euler type scheme for the system of decoupled forward-backward variational...