Weak Euler Approximation for Itô Diffusion and Jump Processes

Abstract

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backwardKolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.

Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients

“…Our approach permits to establish that this bound holds true, up to an additional slowly varying factor in the exponent, for the difference of the densities itself, which again corresponds to the weak error (1.3) for a δ-function. We also mention the recent work of Mikulevičius et al [Mik12], [MZ15], concerning some extensions of [MP91] to jump-driven SDEs with Hölder coefficients.…”

Weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with non-regular drift

“…Using the regularity of the solution of the PDE ∂ t u = Lu with u(0, x) = f (x) where L is the infinitesimal generator of X with coefficients in some Hölder space and F a class of Hölder continuous functions, the weak rate of convergence was given in R. Mikulevičius and E. Platen [37] and R. Mikulevičius and C. Zhang [36]. More precisely, if for α ∈ (2, 3).…”

The Girsanov Theorem Without (So Much) Stochastic Analysis

“…∈ C 4 (R , R )) to achieve a rate 1 [49]. In [38,39], it is shown that for -Hölder continuous coefficients with < 2, the order of convergence is /2. This approach excludes the integer values of , and the terminal condition is required to be (2+ )-Hölder continuous.…”