Pathwise approximation of SDEs by coupling piecewise abelian rough paths

2015/05/06 by Guy Flint, Terry Lyons, Flint, Guy +1
Economics, Econometrics and Finance · Engineering · #FOS: Mathematics #Financial Risk and Volatility Modeling #Fluid Dynamics and Turbulent Flows #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1505.01298

openalex publication_date 2015/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a new pathwise approximation scheme for stochastic differential equations driven by multidimensional Brownian motion which does not require the simulation of Lévy area and has a Wasserstein convergence rate better than the Euler scheme's strong error rate of O(√(h)), where h is the step-size. By using rough path theory we avoid imposing any non-degenerate Hörmander or ellipticity assumptions on the vector fields of the SDE, in contrast to the similar papers of Alfonsi, Davie, and Malliavin et al. The scheme is based on the log-ODE method with the Lévy area increments replaced by Gaussian approximations with the same covariance structure. The Wasserstein coupling is achieved by making small changes to the argument of Davie, the latter being an extension of the Komlós-Major-Tusnády Theorem. We prove that the convergence of the scheme in the Wasserstein metric is of the order O(h^1-2/γ-ε) when the vector fields are γ-Lipschitz in the sense of Stein.

Citations

• Rough differential equation in Banach space driven by weak geometric p-rough path
• Rough paths, Signatures and the modelling of functions on streams
• Approximate Integration of Stochastic Differential Equations
• On the exponential solution of differential equations for a linear operator
• An inequality of the Hölder type, connected with Stieltjes integration

Related

• Pathwise convergence of the Euler scheme for rough and stochastic differential equations
• Wasserstein approximations of the Lévy area random walk via polynomial perturbations of Gaussian distributions
• Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions
• On the pathwise approximation of stochastic differential equations
• Differential equations driven by rough paths: an approach via discrete approximation