Taylor schemes for rough differential equations and fractional diffusions

TL;DR

This paper develops and analyzes two variations of Taylor schemes for rough differential equations and fractional diffusions, focusing on convergence rates and computational efficiency.

Contribution

It introduces incomplete and modified Taylor schemes with explicit error analysis, improving convergence understanding for rough and fractional stochastic differential equations.

Findings

Almost sure convergence rates depend on Hölder exponents.

L_p convergence rates depend on Hurst parameters.

Explicit error functions facilitate convergence analysis.

Abstract

In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the H\"older exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Both rates involves with the incomplete Taylor scheme in a very explicit way and then provide us with the best incomplete schemes, depending on that one needs the almost sure convergence or one needs $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.