Taylor schemes for rough differential equations and fractional diffusions

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.

[1]  Fabrice Baudoin,et al.  Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions , 2011, 1112.2760.

[2]  M. Zähle.  Integration with respect to fractal functions and stochastic calculus. I , 1998 .

[3]  Peter K. Friz,et al.  Multidimensional Stochastic Processes as Rough Paths by Peter K. Friz , 2010 .

[4]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[5]  Terry Lyons,et al.  DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH SIGNALS (I): AN EXTENSION OF AN INEQUALITY OF L. C. YOUNG , 1994 .

[6]  Ivan Nourdin,et al.  Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion , 2006 .

[7]  S. Janson.  Gaussian Hilbert Spaces , 1997 .

[8]  D. Nualart,et al.  Differential equations driven by fractional Brownian motion , 2002 .

[9]  B. Øksendal,et al.  FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE , 2003 .

[10]  Yaozhong Hu.  Multiple integrals and expansion of solutions of differential equations driven by rough paths and by fractional Brownian motions , 2013 .

[11]  Peter K. Friz,et al.  Multidimensional Stochastic Processes as Rough Paths: Theory and Applications , 2010 .

[12]  L. Young.  DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH SIGNALS ( I ) : AN EXTENSION OF AN INEQUALITY OF , 2004 .

[13]  Yaozhong Hu.  Strong and weak order of time discretization schemes of stochastic differential equations , 1996 .

[14]  I︠U︡lii︠a︡ S. Mishura.  Stochastic Calculus for Fractional Brownian Motion and Related Processes , 2008 .

[15]  C. Reutenauer.  Free Lie Algebras , 1993 .

[16]  Ivan Nourdin,et al.  Milstein's type schemes for fractional SDEs , 2007, math/0702317.

[17]  David Nualart,et al.  Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions , 2014, 1408.6471.

[18]  Taylor Expansions and Castell Estimates for Solutions of Stochastic Differential Equations Driven by Rough Paths , 2012, Journal of Stochastic Analysis.