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Partially observed optimal controls of forward-backward doubly stochastic systems

Yufeng Shi, Qingfeng Zhu (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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...

Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection

Sebastian Andres (2009)

Annales de l'I.H.P. Probabilités et statistiques

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...

Path-wise solutions of stochastic differential equations driven by Lévy processes.

David R. E. Williams (2001)

Revista Matemática Iberoamericana

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 for stochastic PDEs

Giuseppe Da Prato (2015)

Banach Center Publications

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

Jean Bertoin (2013)

ESAIM: Probability and Statistics

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

Jean-François Le Gall (2013)

ESAIM: Probability and Statistics

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

Mark Adler (2005)

Annales de l’institut Fourier

In 1962, Dyson showed that the spectrum of a n × n random Hermitian matrix, whose entries (real and imaginary) diffuse according to n 2 independent Ornstein-Uhlenbeck processes, evolves as n non-colliding Brownian particles held together by a drift term. When n → ∞ , 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...

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