Mihai Gradinaru

A singular large deviations phenomenon

Consider {Xte :t⩾0} (e>0), the solution starting from 0 of a stochastic differential equation, which is a small Brownian perturbation of the one-dimensional ordinary differential equation x′t=sgn(xt)|xt|γ (0<γ<1). Denote by pte(x) the density of Xte. We study the exponential decay of the density as e→0. We prove that, for the points (t,x) lying between the extremal solutions of the ordinary differential equation, the rate of the convergence is different from the rate of convergence in large deviations theory (although respected for the points (t,x) which does not lie between the extremals). Proofs are based on probabilistic (large deviations theory) and analytic (viscosity solutions for Hamilton–Jacobi equations) tools.

Milstein's type schemes for fractional SDEs

Weighted power variations of fractional Brownian motion B are used to compute the exact rate of convergence of some approximating schemes associated to one-dimensional stochastic differential equations (SDEs) driven by B. The limit of the error between the exact solution and the considered scheme is computed explicitly.

Abel transform and integrals of Bessel local times

Abstract We study integrals of the type ∫ 0 t ϕ(s) d L s , where ϕ is a positive locally bounded Borel function and L t denotes the local time at level 0 of a Bessel process of dimension d , 0 .

Local Skorokhod topologyon the space of cadlag processes

We modify the global Skorokhod topology, on the space of cadlag paths, by localising with respect to space variable, in order to include the eventual explosions. The tightness of families of probability measures on the paths space endowed with this local Skorokhod topology is studied and a characterization of Aldous type is obtained. The local and global Skorokhod topologies are compared by using a time change transformation. A number of results in the paper should play an important role when studying Levy-type processes with unbounded coefficients by martingale problem approach.

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Let us consider a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient

ON HOMOGENEOUS PINNING MODELS AND PENALIZATIONS

b(t,x)=\rho\,{\rm sgn}(x)|x|^\alpha/t^\beta

Singularities of Hypoelliptic Green Functions

. This process can be viewed as a distorted Brownian motion in a potential, possibly singular, depending on time. After obtaining results on existence and uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters

m-order integrals and generalized Ito's formula; the case of a fractional Brownian motion with any Hurst index

\rho,\alpha

Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H>=1/4

and

Approximation at First and Second Order of m-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales

\beta