A Family of non-Gaussian Martingales with Gaussian Marginals

We construct a family of non-Gaussian martingales the marginals of which are all Gaussian. We give the predictable quadratic variation of these processes and show they do not have continuous paths. These processes are Markovian and inhomogeneous in time, and we give their infinitesimal generators. Within this family we find a class of piecewise deterministic pure jump processes and describe the laws of jumps and times between the jumps.Comment: 16 pages, 2 figure

Branching Processes: Their Role in Epidemiology

Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large

A new stochastic differential equation approach for waves in a random medium

We present a mathematical approach that simplifies the theoretical treatment of electromagnetic localization in random media and leads to closed form analytical solutions. Starting with the assumption that the dielectric permittivity of the medium has delta-correlated spatial fluctuations, and using the Ito lemma, we derive a linear stochastic differential equation for a one dimensional random medium. The equation leads to localized wave solutions. The localized wave solutions have a localization length that scales inversely with the square of the frequency of the wave in the low frequency regime, whereas in the high frequency regime, this length varies inversely with the frequency to the power of two thirds