The Annals of Statistics

ANOVA for diffusions and Itô processes

Per Aslak Mykland and Lan Zhang

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Itô processes are the most common form of continuous semimartingales, and include diffusion processes. This paper is concerned with the nonparametric regression relationship between two such Itô processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a unit of time (such as a day). A main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed, the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term.

The proposed methodology, “ANOVA for diffusions and Itô processes,” can be used to measure the statistical quality of a parametric model and, nonparametrically, the appropriateness of a one-regressor model in general. On the other hand, it also helps quantify and characterize the trading (hedging) error in the case of financial applications.

Article information

Ann. Statist. Volume 34, Number 4 (2006), 1931-1963.

First available in Project Euclid: 3 November 2006

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Mathematical Reviews number (MathSciNet)

Primary: 60G44: Martingales with continuous parameter 62M09: Non-Markovian processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 91B28
Secondary: 60G42: Martingales with discrete parameter 62G20: Asymptotic properties 62P20: Applications to economics [See also 91Bxx] 91B84: Economic time series analysis [See also 62M10]

ANOVA continuous semimartingale statistical uncertainty goodness of fit discrete sampling parametric and nonparametric estimation small interval asymptotics stable convergence option hedging


Mykland, Per Aslak; Zhang, Lan. ANOVA for diffusions and Itô processes. Ann. Statist. 34 (2006), no. 4, 1931--1963. doi:10.1214/009053606000000452.

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