Stability of densities for perturbed Diffusions and Markov Chains

We are interested in studying the sensitivity of diffusion processes or their approximations by Markov Chains with respect to a perturbation of the coefficients.Comment: 26 page

Weak Error for stable driven SDEs: expansion of the densities

Consider a multidimensional SDE of the form $X_t = x+\int_{0}^{t} b(X_{s-})ds+\int{0}^{t} f(X_{s-})dZ_s$ where $(Z_s)_{s\ge 0}$ is a symmetric stable process. Under suitable assumptions on the coefficients the unique strong solution of the above equation admits a density w.r.t. the Lebesgue measure and so does its Euler scheme. Using a parametrix approach, we derive an error expansion at order 1 w.r.t. the time step for the difference of these densities.Comment: 27 page

A probabilistic interpretation of the parametrix method

In this article, we introduce the parametrix technique in order to construct fundamental solutions as a general method based on semigroups and their generators. This leads to a probabilistic interpretation of the parametrix method that is amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with H\"{o}lder continuous coefficients.Comment: Published at http://dx.doi.org/10.1214/14-AAP1068 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On some Non Asymptotic Bounds for the Euler Scheme

We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration is derived from a Gaussian upper bound of the density of the scheme and a modification of the so-called "Herbst argument" used to prove Logarithmic Sobolev inequalities. We eventually establish a Gaussian lower bound for the density of the scheme that emphasizes the concentration is sharp.Comment: 26 page