Weak error on the densities for the Euler scheme of stable additive SDEs with H{ö}lder drift
This provides a theoretical convergence rate for density approximation in a class of jump-driven SDEs with irregular drift, which is incremental for the numerical analysis of stochastic differential equations.
The authors prove that the weak error on densities for the randomized Euler-Maruyama scheme of stable additive SDEs with Hölder drift converges at rate (α+β-1)/α, where α∈(1,2] is the stability index and β∈(0,1) is the Hölder exponent of the 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 $α$-stable process, $α$ $\in$ (1, 2] and the drift b $\in$ L$\infty$ ([0,T],C$β$(Rd,Rd)), $β$ $\in$ (0,1), is bounded and H{ö}lder regular in space. Using an Euler scheme with a randomization of the time variable, we show that, denoting $γ$\,:= $α$ + $β$ -- 1, the weak error on densities related to this discretization converges at the rate $γ$/$α$.