Asymptotic Error Distributions for Euler-Maruyama Schemes with Additive Semimartingale Noise

Abstract

We establish a joint stable limit theorem for integrated discretization errors of finite-dimensional càdlàg semimartingales whose predictable characteristics are absolutely continuous in time and satisfy a pathwise integrability condition. The normalized integrated error, evaluated on the discretization grid, converges stably jointly with the discretized semimartingale in the vector-valued Skorokhod $J_1$-topology. No martingale representation property is assumed. We apply this theorem to the Euler-Maruyama scheme for stochastic differential equations with additive semimartingale noise. For a $C^1$ drift with locally Lipschitz derivative and a nonexplosive solution, the grid-point error normalized by $n$ converges stably jointly with the discretized driver. The limiting error solves a linear integral equation with an explicit inhomogeneous term involving the driver, an auxiliary Brownian motion, and independent uniform marks attached to the jumps. We also obtain the same limit under two alternative drift regularity conditions based on small-jump integrability or Lévy semigroup smoothing. For continuous drivers, the continuous-time error converges stably in the uniform topology.