Weak error analysis via functional It\^o calculus

TL;DR

This paper develops a weak error analysis framework for stochastic differential equations using functional Itô calculus, providing a general error representation and demonstrating a convergence rate of 1 for Euler approximations.

Contribution

It introduces a novel weak error representation formula for path-dependent functionals of SDE solutions using functional Itô calculus, applicable to a broad class of functionals.

Findings

Derived a general weak error representation formula.

Established a weak convergence rate of 1 for Euler approximations.

Demonstrated the applicability in a one-dimensional setting.

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 It\^o calculus, such as the functional It\^o 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 Fr\'echet 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.