arXiv subjects

Mathis Fitoussi

Publications and source records attributed to Mathis Fitoussi.

4 recordsLinked to original sources

arXiv2025

Weak Error on the densities for the Euler scheme of stable additive SDEs with Besov drift

We are interested in the Euler-Maruyama dicretization of the formal SDE, $dX_t=b(t,X_t)dt+dZ_t$, where $Z$ is a symmetric isotropic d dimensional stable process of index $\alpha\in (1,2)$, and $b$ is distributional. It belongs to a mix Lebesgue-Besov space. The associated parameters satisfy some constraints which guarantee weak-well posedness. Defining an appropriate Euler scheme, we obtain a convergence rate for the weak error on the densities. The rate depends on the parameters.

math.AP↗

arXiv2024

Weak error on the densities for the Euler scheme of stable additive SDEs with H{\"o}lder drift

We are interested in the Euler-Maruyama dicretization of the SDE dXt =b(t,Xt)dt+ dZt, X0 =x$\in$Rd, where Zt is a symmetric isotropic d-dimensional $\alpha$-stable process, $\alpha$ $\in$ (1, 2] and the drift b $\in$ L$\infty$ ([0,T],C$\beta$(Rd,Rd)), $\beta$ $\in$ (0,1), is bounded and H{\"o}lder regular in space. Using an Euler scheme with a randomization of the time variable, we show that, denoting $\gamma$\,:= $\alpha$ + $\beta$ -- 1, the weak error on densities related to this discretization converges at the rate $\gamma$/$\alpha$.

math.NA↗

arXiv2024

Weak well-posedness and weak discretization error for stable-driven SDEs with Lebesgue drift

We are interested in the discretization of stable driven SDEs with additive noise for $\alpha$ $\in$ (1, 2) and Lq -- Lp drift under the Serrin type condition $\alpha$/q + d/p < $\alpha$ -- 1. We show weak existence and uniqueness as well as heat kernel estimates for the SDE and obtain a convergence rate of order (1/$\alpha$)*($\alpha$ -- 1 -- $\alpha$/q - d/p) for the difference of the densities for the Euler scheme approximation involving suitably cutoffed and time randomized drifts.

math.PR↗

arXiv2023

Heat kernel estimates for stable-driven SDEs with distributional drift

We consider the formal SDE dX t = b(t, X t)dt + dZ t , X 0 = x $\in$ R d , (E) where b $\in$ L r ([0, T ], B $\beta$ p,q (R d , R d)) is a time-inhomogeneous Besov drift and Z t is a symmetric d-dimensional $\alpha$-stable process, $\alpha$ $\in$ (1, 2), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, L r and B $\beta$ p,q respectively denote Lebesgue and Besov spaces. We show that, when $\beta$ > (1--$\alpha$+ $\alpha$/r + d/p)/2 , the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using Duhamel expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

math.PR↗