Weak error analysis via functional Itô calculus

Abstract

We consider autonomous stochastic ordinary differential equations (SDEs) and weak approximations of their solutions for a general class of sufficiently smooth path-dependent functionals f. Based on tools from functional Itô calculus, such as the functional Itô formula and functional Kolmogorov equation, we derive a general representation formula for the weak error $E(f(X_T)-f({\tilde{X}}_T))$, where $X_T$ and ${\tilde{X}}_T$ are the paths of the solution process and its approximation up to time T. The functional $f:C([0,T],R^d)\to R$ is assumed to be twice continuously Fréchet differentiable with derivatives of polynomial growth. The usefulness of the formula is demonstrated in the one dimensional setting by showing that if the solution to the SDE is approximated via the linearly time-interpolated explicit Euler method, then the rate of weak convergence for sufficiently regular f is 1.

arXiv comment35 pages. Fixed several typos added some more references and relaxed the assumptions of Theorem 8.1

References in corpus (4)

• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Weak error analysis for semilinear stochastic Volterra equations with additive noise
• Weak approximation of martingale representations
• Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization