Preprint
Weak error analysis via functional Itô calculus
Publication Details
Authors: | Lindner, F.; Kovács, M. |
Publication year: | 2016 |
Journal: | arXiv Preprint |
Pages range : | 1-35 |
Journal acronym: | arXiv |
DOI-Link der Erstveröffentlichung: |
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{ô} calculus, such as the functional It{ô} formula and functional Kolmogorov equation, we derive a general representation formula for the weak error {\$}E(f(X{\_}T)-f($\backslash$tilde X{\_}T)){\$}, where $X_T$ and {\$}$\backslash$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)$\backslash$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. 35 pages. Fixed several typos added some more references and relaxed the assumptions of Theorem 8.1
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{ô} calculus, such as the functional It{ô} formula and functional Kolmogorov equation, we derive a general representation formula for the weak error {\$}E(f(X{\_}T)-f($\backslash$tilde X{\_}T)){\$}, where $X_T$ and {\$}$\backslash$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)$\backslash$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. 35 pages. Fixed several typos added some more references and relaxed the assumptions of Theorem 8.1
