ANOVA for diffusions and Itô processes

Ito 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 Ito 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 Ito processes," can be used to measure the statistical quality of a parametric model and, non-parametrically, 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.

[1]  K. E. Dambis,et al.  On the Decomposition of Continuous Submartingales , 1965 .

[2]  L. Dubins,et al.  ON CONTINUOUS MARTINGALES. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[4]  David Aldous,et al.  On Mixing and Stability of Limit Theorems , 1978 .

[5]  B. Efron,et al.  Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information , 1978 .

[6]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[7]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[8]  D. Sondermann.  Hedging of non-redundant contingent claims , 1985 .

[9]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[10]  P. Protter.  Stochastic integration and differential equations , 1990 .

[11]  Risk-Minimality and Orthogonality of Martingales , 1990 .

[12]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[13]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[14]  Asymptotic Expansions for Martingales , 1993 .

[15]  D. Florens-zmirou.  On estimating the diffusion coefficient from discrete observations , 1993, Journal of Applied Probability.

[16]  J. Jacod,et al.  Estimation of the diffusion coefficient for diffusion processes: random sampling , 1994 .

[17]  Dean P. Foster,et al.  Continuous Record Asymptotics for Rolling Sample Variance Estimators , 1994 .

[18]  P. Mykland.  Embedding and asymptotic expansions for martingales , 1995 .

[19]  Walter Schachermayer,et al.  The Existence of Absolutely Continuous Local Martingale Measures (1995) , 1995 .

[20]  Walter Schachermayer,et al.  The no-arbitrage property under a change of numéraire , 1995 .

[21]  N. Shephard,et al.  Likelihood INference for Discretely Observed Non-linear Diffusions , 2001 .

[22]  P. Protter,et al.  Asymptotic error distributions for the Euler method for stochastic differential equations , 1998 .

[23]  M. Hofmann.  Lp estimation of the diffusion coefficient , 1999 .

[24]  Thierry Jeantheau,et al.  Parameter estimation for discretely observed stochastic volatility models , 1999 .

[25]  C. LareÂdo,et al.  Stochastic volatility models as hidden Markov models and statistical applications , 2000 .

[26]  Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient , 2000 .

[27]  Non‐parametric Kernel Estimation of the Coefficient of a Diffusion , 2000 .

[28]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[29]  Lan Zhang,et al.  From martingales to ANOVA : implied and realized volatility , 2001 .

[30]  Martin Jacobsen.  Discretely Observed Diffusions: Classes of Estimating Functions and Small Δ‐optimality , 2001 .

[31]  H. Sørensen.  Discretely Observed Diffusions: Approximation of the Continuous‐time Score Function , 2001 .

[32]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[33]  M. Hoffmann.  Rate of convergence for parametric estimation in a stochastic volatility model , 2002 .

[34]  J. Hull.  Options, futures & other derivatives , 2003 .

[35]  Jianqing Fan.  Rejoinder: A selective overview of nonparametric methods in financial econometrics , 2004, math/0411034.

[36]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[37]  Lan Zhang.  Efficient Estimation of Stochastic Volatility Using Noisy Observations: A Multi-Scale Approach , 2004, math/0411397.

[38]  Zhou Zhou,et al.  “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data” , 2005 .

[39]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .