Error Distribution of the Local Linearization Method for Stochastic Differential Equations with Additive Brownian Noise

Abstract

We prove a functional stable limit theorem for the discretization error process of a local linearization scheme for stochastic differential equations with additive Brownian noise. The scheme includes the conditional mean of the second-order term involving the Brownian increment in the Taylor expansion of the drift. The leading error is then formed by centered quadratic terms in the Brownian increments, and the sharp normalization is \(n\sqrt n\). Under \(C^3\)-regularity and a Lyapunov-type condition on the drift, the scaled error process converges stably in \(C([0,1],\mathbb R^d)\) to the solution of the limiting linear stochastic differential equation. The martingale part of the limit is driven by a Brownian motion independent of the original \(\sigma\)-field, and its coefficient is determined by the Hessian of the drift and the covariance matrix of the additive noise.