International audienceWe investigate branching properties of the solution of a stochastic differential equation of fragmentation (SDEF) and we properly associate a continuous time càdlàg Markov process on the space S# of all fragmentation sizes, introduced by J. Bertoin. A binary fragmentation kernel induces a specific class of integral type branching kernels and taking as base process the solution of the initial (SDEF), we construct a branching process corresponding to a rate of loss of mass greater than a given strictly positive size d. It turns out that this branching process takes values in the set of all finite configurations of sizes greater than d. The process on S# is then obtained by letting d tend to zero. A key argument for the convergence of the branching processes is given by the Bochner-Kolmogorov theorem. The construction and the proof of the path regularity of the Markov processes are based on several newly developed potential theoretical tools, in terms of excessive functions and measures, compact Lyapunov functions, and some appropriate absorbing sets

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that exists between Cox-Ingersoll-Ross processes and Bessel processes.

In this article, we introduce an algorithm that simulates efficiently the first exit time and position from a rectangle (or a parallelepiped) for a Brownian motion that starts at any point inside. This method provides an exact way to simulate the first exit time and position from any polygonal domain and then to solve some Dirichlet problems, whatever the dimension. This method can be used as a replacement or complement of the method of the random walk on spheres and can be easily adapted to deal with Neumann boundary conditions or Brownian motion with a constant drift.