In this article, we generalize the lower bound estimates for uniformly elliptic diffusion processes obtained by Kusuoka and Stroock. We define the concept of uniform elliptic random variable on Wiener space and show that with this definition one can prove a lower bound estimate of Gaussian type for its density. We apply our results to the case of the stochastic heat equation under the hypothesis of unifom ellipticity of the diffusion coefficient.

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 Hölder continuous coefficients.

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with N steps is smaller than O(N −2/3+ε ) where ε is an arbitrary positive constant. This rate is intermediate between the strong error estimation in O(N −1/2 ) obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation O(N −1 ) obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time T . We also check that the supremum over t ∈ [0, T ] of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time t and the Euler scheme at time t behaves like O( log(N )N −1 ). 0. Introduction. For σ : R → R and b : R → R, we are interested in the simulation of the stochastic differential equationwhere X 0 = x 0 ∈ R and W = (W t ) t≥0 is a standard Brownian motion. We make the standard Lipschitz assumptions on the coefficients, ∃K ∈ (0, +∞), ∀x, y ∈ R |σ(x) − σ(y)| + |b(x) − b(y)| ≤ K|x − y|.

In this article, we consider an unbiased simulation method for multidimensional diffusions based on the parametrix method for solving partial differential equations with Hölder continuous coefficients. This Monte Carlo method which is based on an Euler scheme with random time steps, can be considered as an infinite dimensional extension of the Multilevel Monte Carlo method for solutions of stochastic differential equations with Hölder continuous coefficients. In particular, we study the properties of the variance of the proposed method. In most cases, the method has infinite variance and therefore we propose an importance sampling method to resolve this issue.

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