Asymptotic Error Distributions of the Crank-Nicholson Scheme for SDEs Driven by Fractional Brownian Motion

We investigate the difference between the solution to a stochastic differential equation driven by a fractional Brownian motion and the approximation by the Crank-Nicholson scheme associated with the equation. In preceding results, researchers deal with the errors of the Euler scheme and the Crank-Nicholson scheme for some fixed time as real-valued random variables and study the convergence rates and the limit distributions. In the present paper, we consider the error as stochastic processes and determine the convergence rate of the error and the limit distribution in the Skorohod topology. The limit distribution is expressed in terms of the solution to the equation and the It integral with respect to a standard Brownian motion independent of the driving process of the equation. This result extends those contained in J Theor Probab 20(4):871-899, 2007. The key ingredients in our proof are asymptotic behavior of weighted Hermite variations as stochastic processes. We also give the It formula for fractional Brownian motion.