We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density...

We investigate solutions of backward stochastic differential equations (BSDEs) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Lévy process. In these new types of equations, the generator can depend on the past values...

In this paper, we prove the validity of the Edgeworth expansion of the Discrete Fourier transforms of some linear time series. This result is applied to approach moments of non-linear functionals of the periodogram. As an illustration, we give an expression of the mean square error of the...

We prove results on the existence of Doléans-Dade measures and of the Doob-Meyer decomposition for supermartingales indexed by a general index set.

We study the sample path regularity of a second-order random field (Xt)t[set membership, variant]T where T is an open subset of . It is shown that the conditions on its covariance function, known to be equivalent to mean square differentiability of order k, imply that the sample paths are...

We study critical branching random walks (BRWs) U(n) on  where the displacement of an offspring from its parent has drift  towards the origin and reflection at the origin. We prove that for any [alpha]1, conditional on survival to generation [n[alpha]], the maximal displacement is ....

We study models of discrete-time, symmetric, -valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances [omega]xy[set membership, variant][0,1], with polynomial tail near 0 with exponent [gamma]0. We first prove for all d=5 that the return...

We study the adaptive dynamics of predator-prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long...

We present new algorithms for weak approximation of stochastic differential equations driven by pure jump Lévy processes. The method uses adaptive non-uniform discretization based on the times of large jumps of the driving process. To approximate the solution between these times we replace the...

We trace Itô's early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô's formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô's formula in mathematical...