In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work (Stochastic Process. Appl. 123 (2013) 887–933), which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.
References
[1] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331. 0376.60026 10.1214/aop/1176995577 euclid.aop/1176995577[1] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331. 0376.60026 10.1214/aop/1176995577 euclid.aop/1176995577
[3] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 43–60. 0838.60051 10.1007/BF01303802[3] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 43–60. 0838.60051 10.1007/BF01303802
[4] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 93–128. 0866.60049 10.1515/mcma.1996.2.2.93[4] Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 93–128. 0866.60049 10.1515/mcma.1996.2.2.93
[5] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M. and Shephard, N. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance 33–68. Springer, Berlin. 1106.60037[5] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M. and Shephard, N. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance 33–68. Springer, Berlin. 1106.60037
[6] Chan, K. S. and Stramer, O. (1998). Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients. Stochastic Process. Appl. 76 33–44. MR1638015 0934.60052 10.1016/S0304-4149(98)00020-9[6] Chan, K. S. and Stramer, O. (1998). Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients. Stochastic Process. Appl. 76 33–44. MR1638015 0934.60052 10.1016/S0304-4149(98)00020-9
[7] Dalalyan, A. and Yoshida, N. (2011). Second-order asymptotic expansion for a non-synchronous covariation estimator. Ann. Inst. Henri Poincaré Probab. Stat. 47 748–789. 1328.62511 10.1214/10-AIHP383 euclid.aihp/1308834858[7] Dalalyan, A. and Yoshida, N. (2011). Second-order asymptotic expansion for a non-synchronous covariation estimator. Ann. Inst. Henri Poincaré Probab. Stat. 47 748–789. 1328.62511 10.1214/10-AIHP383 euclid.aihp/1308834858
[8] Halidias, N. and Kloeden, P. E. (2006). A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient. J. Appl. Math. Stoch. Anal. Art. ID 73257. 1118.60051 10.1155/JAMSA/2006/73257[8] Halidias, N. and Kloeden, P. E. (2006). A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient. J. Appl. Math. Stoch. Anal. Art. ID 73257. 1118.60051 10.1155/JAMSA/2006/73257
[9] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam; Kodansha, Ltd., Tokyo.[9] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam; Kodansha, Ltd., Tokyo.
[10] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307. 0937.60060 10.1214/aop/1022855419 euclid.aop/1022855419[10] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307. 0937.60060 10.1214/aop/1022855419 euclid.aop/1022855419
[11] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. 1018.60002[11] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. 1018.60002
[12] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin. 0752.60043[12] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin. 0752.60043
[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. MR1119837 0762.60047[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. MR1119837 0762.60047
[14] Mikulevicius, R. and Platen, E. (1991). Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151 233–239. 0733.65104 10.1002/mana.19911510114[14] Mikulevicius, R. and Platen, E. (1991). Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151 233–239. 0733.65104 10.1002/mana.19911510114
[15] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Springer, Berlin. 1099.60003[15] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Springer, Berlin. 1099.60003
[16] Podolskij, M., Veliyev, B. and Yoshida, N. (2017). Edgeworth expansion for the pre-averaging estimator. Stochastic Process. Appl. 127 3558–3595. 1381.60071 10.1016/j.spa.2017.03.001[16] Podolskij, M., Veliyev, B. and Yoshida, N. (2017). Edgeworth expansion for the pre-averaging estimator. Stochastic Process. Appl. 127 3558–3595. 1381.60071 10.1016/j.spa.2017.03.001
[17] Podolskij, M. and Yoshida, N. (2016). Edgeworth expansion for functionals of continuous diffusion processes. Ann. Appl. Probab. 26 3415–3455. 1417.62238 10.1214/16-AAP1179 euclid.aoap/1481792589[17] Podolskij, M. and Yoshida, N. (2016). Edgeworth expansion for functionals of continuous diffusion processes. Ann. Appl. Probab. 26 3415–3455. 1417.62238 10.1214/16-AAP1179 euclid.aoap/1481792589
[19] Yan, L. (2002). The Euler scheme with irregular coefficients. Ann. Probab. 30 1172–1194. MR1920104 1020.60054 10.1214/aop/1029867124 euclid.aop/1029867124[19] Yan, L. (2002). The Euler scheme with irregular coefficients. Ann. Probab. 30 1172–1194. MR1920104 1020.60054 10.1214/aop/1029867124 euclid.aop/1029867124
[20] Yoshida, N. (1997). Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Related Fields 109 301–342. 0888.60020 10.1007/s004400050134[20] Yoshida, N. (1997). Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Related Fields 109 301–342. 0888.60020 10.1007/s004400050134
[21] Yoshida, N. (2012). Asymptotic expansion for the quadratic form of the diffusion process. Working paper. Available at arXiv:1212.5845. 1212.5845[21] Yoshida, N. (2012). Asymptotic expansion for the quadratic form of the diffusion process. Working paper. Available at arXiv:1212.5845. 1212.5845
[22] Yoshida, N. (2013). Martingale expansion in mixed normal limit. Stochastic Process. Appl. 123 887–933. 1261.60034 10.1016/j.spa.2012.10.007[22] Yoshida, N. (2013). Martingale expansion in mixed normal limit. Stochastic Process. Appl. 123 887–933. 1261.60034 10.1016/j.spa.2012.10.007