## The Annals of Applied Probability

### Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

#### Abstract

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_{n}^{-1}$, where $\gamma_{n}=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $\gamma_{n}=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $\gamma_{n}=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_{t},0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_{t}^{n},0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $\gamma_{n}(X^{n}-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^{p}$ convergence of $n(X^{n}_{t}-X_{t})$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 1147-1207.

Dates
First available in Project Euclid: 22 March 2016

https://projecteuclid.org/euclid.aoap/1458651830

Digital Object Identifier
doi:10.1214/15-AAP1114

Mathematical Reviews number (MathSciNet)
MR3476635

Zentralblatt MATH identifier
1339.60095

#### Citation

Hu, Yaozhong; Liu, Yanghui; Nualart, David. Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions. Ann. Appl. Probab. 26 (2016), no. 2, 1147--1207. doi:10.1214/15-AAP1114. https://projecteuclid.org/euclid.aoap/1458651830

#### References

• [1] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331.
• [2] Cambanis, S. and Hu, Y. (1996). Exact convergence rate of the Euler–Maruyama scheme, with application to sampling design. Stochastics Stochastics Rep. 59 211–240.
• [3] Corcuera, J. M., Nualart, D. and Podolskij, M. (2014). Asymptotics of weighted random sums. Commun. Appl. Ind. Math. 6 e–486, 11.
• [4] Deya, A., Neuenkirch, A. and Tindel, S. (2012). A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 518–550.
• [5] Friz, P. and Riedel, S. (2014). Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 154–194.
• [6] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
• [7] Hu, Y. (1996). Strong and weak order of time discretization schemes of stochastic differential equations. In Séminaire de Probabilités, XXX. Lecture Notes in Math. 1626 218–227. Springer, Berlin.
• [8] Hu, Y. and Nualart, D. (2007). Differential equations driven by Hölder continuous functions of order greater than $1/2$. In Stochastic Analysis and Applications. Abel Symp. 2 399–413. Springer, Berlin.
• [9] Hu, Y. and Nualart, D. (2009). Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 285–328.
• [10] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
• [11] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
• [12] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
• [13] Kurtz, T. G. and Protter, P. (1991). Wong–Zakai corrections, random evolutions, and simulation schemes for SDEs. In Stochastic Analysis 331–346. Academic Press, Boston, MA.
• [14] Lyons, T. (1994). Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett. 1 451–464.
• [15] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Univ. Press, Oxford.
• [16] Mémin, J., Mishura, Y. and Valkeila, E. (2001). Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51 197–206.
• [17] Mishura, Y. S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math. 1929. Springer, Berlin.
• [18] Neuenkirch, A. and Nourdin, I. (2007). Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J. Theoret. Probab. 20 871–899.
• [19] Neuenkirch, A., Tindel, S. and Unterberger, J. (2010). Discretizing the fractional Lévy area. Stochastic Process. Appl. 120 223–254.
• [20] Nourdin, I., Nualart, D. and Tudor, C. A. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46 1055–1079.
• [21] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
• [22] Nualart, D. (2003). Stochastic integration with respect to fractional Brownian motion and applications. In Stochastic Models (Mexico City, 2002). Contemp. Math. 336 3–39. Amer. Math. Soc., Providence, RI.
• [23] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
• [24] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
• [25] Nualart, D. and Răşcanu, A. (2002). Differential equations driven by fractional Brownian motion. Collect. Math. 53 55–81.
• [26] Nualart, D. and Saussereau, B. (2009). Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 391–409.
• [27] Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247–262. Springer, Berlin.
• [28] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251–291.
• [29] Pipiras, V. and Taqqu, M. S. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7 873–897.
• [30] Rosenblatt, M. (2011). Selected Works of Murray Rosenblatt. Springer, New York.
• [31] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509.
• [32] Tudor, C. A. (2008). Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12 230–257.
• [33] Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 251–282.
• [34] Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 333–374.