stability of solutions of symmetric stochastic differential equations with discontinuous driving semimartingales
Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients
We prove the unique solvability of parabolic equations with discontinuous leading coefficients in . Using this result, we establish the uniqueness of diffusion processes with time-dependent discontinuous coefficients.
Second-order neutral stochastic evolution equations with heredity.
Self attracting diffusions: two case studies.
Self-interacting diffusions II : convergence in law
Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab. 15 (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric limit measure associated...
Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab. 15 (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there...
Semiclassical analysis and a new result for Poisson-Lev́y excursion measures.
Série de Taylor stochastique et formule de Campbell-Hausdorff, d'après Benarous
Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...
Sharp estimates for the convergence of the density of the Euler scheme in small time.
Simplified Malliavin calculus
Simulation and approximation of Lévy-driven stochastic differential equations
We consider the approximate Euler scheme for Lévy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Lévy measure of the driving process behaves like |z|−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the...
Simulation and approximation of Lévy-driven stochastic differential equations
We consider the approximate Euler scheme for Lévy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Lévy measure of the driving process behaves like |z|−1−αdz near 0, for some α∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the...
Singularity functions for fractional processes : application to the fractional brownian sheet
Skew products, regular conditional probabilities and stochastic differential equations : a technical remark
Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump.
Solution explicite de l’équation
Solutions faibles et semi-martingales
Solutions of stochastic differential equations obeying the law of the iterated logarithm, with applications to financial markets.