arXiv subjects

Pierre Etore

Publications and source records attributed to Pierre Etore.

7 recordsLinked to original sources

arXiv2022

Stochastic processes associated to multidimensional parabolic transmission problems in divergence form

In this note we define and study the stochastic process $X$ in link with a parabolic transmission operator $(A,D(A))$ in divergence form. The transmission operator involves a diffraction condition along a transmission boundary. To that aim we gather and clarify some results coming from the theory of Dirichlet forms as exposed in [6] and [14] for general divergence form operators. We show that $X$ is a semimartingale and that it is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary.

math.AP↗

arXiv2020

A transformed stochastic Euler scheme for multidimensional transmission PDE

In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is assumed to present a compatibility transmission condition of its conormal derivatives at this interface (multi-dimensional diffraction problem). We prove an existence and uniqueness result for the solution and construct a low complexity numerical Monte Carlo stochastic Euler scheme to approximate the solution of the parabolic partial differential equation in divergence form. In particular, we give new estimates for the partial derivatives of the solution. Using these estimates, we prove a convergence rate for our stochastic numerical method when the initial condition belongs to an iterated domain of the divergence form operator. Our method presents the same convergence rate as the stochastic numerical schemes elaborated for the same problem in the one-dimensional context. Finally, we compare our results to classical deterministic numerical approximations and illustrate the accuracy of our method.

math.PR↗

arXiv2014

L_1-distance for additive processes with time-homogeneous Lévy measures

We give an explicit bound for the $L_1$-distance between two additive processes of local characteristics $(f_j(\cdot),σ^2(\cdot),ν_j)$, $j = 1,2$. The cases $σ=0$ and $σ> 0$ are both treated. We allow $ν_1$ and $ν_2$ to be equivalent time-homogeneous Lévy measures, possibly with infinite variation. Some examples of possible applications are discussed.

math.PR↗

arXiv2013

Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift

In this note we propose an exact simulation algorithm for the solution of dX_t=dW_t+b(X_t)dt (1) where b is a smooth real function except at point 0 where b(0+)\neq b(0-). The main idea is to sample an exact skeleton of X using an algorithm deduced from the convergence of the solutions of the skew perturbed equation dX^β_t=dW_t+b(X^β_t)dt + βdL^0_t {X^β} (2) towards X solution of (1) as βtends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in \cite{etoremartinez1} for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as βtends to 0, we prove that this limit algorithm is an exact simulation algorithm for the solution of the limit equation (1). Numerical examples are shown to illustrate the performance of this exact simulation algorithm.

math.PR↗

arXiv2013

Exact Simulation of One-dimensional Stochastic Differential Equations involving the local time at zero of the unknown process

In this article we extend the exact simulation methods of Beskos et al. to the solutions of one-dimensional stochastic differential equations involving the local time of the unknown process at point zero. In order to perform the method we compute the law of the skew Brownian motion with drift. The method presented in this article covers the case where the solution of the SDE with local time corresponds to a divergence form operator with a discontinuous coefficient at zero. Numerical examples are shown to illustrate the method and the performances are compared with more traditional discretization schemes.

math.PR↗

arXiv2012

On the existence of a time inhomogeneous skew Brownian motion and some related laws

This article is devoted to the construction of a solution for the "skew inhomogeneous Brownian motion" equation, which first appear in a seminal paper by Sophie Weinryb (1983). We investigate some laws related to the constructed process. In particular, using the description of the straddling excursion above a deterministic time, we compute the joint law of the process, its local time and its straddling time.

math.PR↗

arXiv2007

Adaptive optimal allocation in stratified sampling methods

In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm.

stat.ME↗