Partial Sums of Orthonormal Bases Preserving Positivity - and Martingales.
Partially observed optimal controls of forward-backward doubly stochastic systems
The partially observed optimal control problem is considered for forward-backward doubly stochastic systems with controls entering into the diffusion and the observation. The maximum principle is proven for the partially observable optimal control problems. A probabilistic approach is used, and the adjoint processes are characterized as solutions of related forward-backward doubly stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed...
Partition-dependent stochastic measures and -deformed cumulants.
Past and Future
Path continuity and last exit distributions
Path decompositions for real Levy processes
Path properties of a class of locally asymptotically self similar processes.
Path transformations of first passage bridges.
Paths of finitely additive brownian motion need not be bizarre
Pathwise approximations of processes based on the fine structure of their filtrations
Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection
In this paper, the object of study is a Skorohod SDE in a convex polyhedron with oblique reflection at the boundary. We prove that the solution is pathwise differentiable with respect to its deterministic starting point up to the time when two of the faces are hit simultaneously. The resulting derivatives evolve according to an ordinary differential equation, when the process is in the interior of the polyhedron, and they are projected to the tangent space, when the process hits the boundary, while...
Pathwise differentiability with respect to a parameter of solutions of stochastic differential equations
Path-wise solutions of stochastic differential equations driven by Lévy processes.
In this paper we show that a path-wise solution to the following integral equationYt = ∫0t f(Yt) dXt, Y0 = a ∈ Rd,exists under the assumption that Xt is a Lévy process of finite p-variation for some p ≥ 1 and that f is an α-Lipschitz function for some α > p. We examine two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric, the other forward. The geometric solution is obtained by adding fictitious time and solving an associated...
Pathwise uniqueness and approximation of solutions of stochastic differential equations
Pathwise uniqueness for reflecting Brownian motion in certain planar Lipschitz domains.
Pathwise uniqueness for stochastic PDEs
We consider a stochastic evolution equation in a separable Hilbert spaces H or in a separable Banach space E with a Hölder continuous perturbation on the drift. We review some recent result about pathwise uniqueness for this equation.
Paul Lévy et l’arithmétique des lois de probabilités
Ce court texte reprend un exposé donné le 15 Décembre 2011 au Laboratoire de Probabilités et Modèles Aléatoires, lors d’une journée en hommage à Paul Lévy. On y rappellera comment des considérations sur l’arithmétique des lois de probabilités ont conduit Lévy à étudier les processus à accroissements indépendants.
Paul Lévy et le mouvement brownien
Ce texte est tiré d’un exposé présenté au cours de la journée Paul Lévy organisée au Laboratoire de Probabilités et Modèles Aléatoires de l’Université Pierre et Marie Curie le 15 décembre 2011. L’objectif de cet exposé était de donner un aperçu des contributions de Paul Lévy à la théorie du mouvement brownien.
PDE's for the Dyson, Airy and Sine processes
In 1962, Dyson showed that the spectrum of a random Hermitian matrix, whose entries (real and imaginary) diffuse according to independent Ornstein-Uhlenbeck processes, evolves as non-colliding Brownian particles held together by a drift term. When , the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend...
PDE-viscosity solution approach to some problems of large deviations