Anova for diffusions and itô processes

Annals of Statistics

Volumn 34, Issue 4, 2006, Pages 1931-1963

  Mykland, Per Aslak ^a   Zhang, Lan ^b,c  

^a UNIVERSITY OF CHICAGO   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

^c UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

ANOVA; Continuous semimartingale; Discrete sampling; Goodness of fit; Option hedging; Parametric and nonparametric estimation; Small interval asymptotics; Stable convergence; Statistical uncertainty

Indexed keywords

EID: 33745653210     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000452     Document Type: Article

References (46)

1. ALDOUS, D. and EAGLESON, O. (1978). On mixing and stability of limit theorems. Ann. Probdb. 6 325-331. MR0517416
2. BARNDORFF-NIELSEN, O. and SHEPHARD, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167-241. MR1841412
3. BARNDORFF-NIELSEN, O. and SHEPHARD, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 253-280. MR1904704
4. BIBBY, B. M., JACOBSEN, M. and SØRENSEN, M. (2006). Estimating functions for discretely sampled diffusion-type models. In Handbook of Financial Econometrics (Y. Aït-Sahalia and L. P. Hansen, eds.). North-Holland, Amsterdam. To appear.
5. BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD. MR1245941
6. COX, J., INGERSOLL, J. and ROSS, S. (1985). A theory of the term structure of interest rates. Econometrica 53 385-407. MR0785475
7. DAMBIS, K. (1965). On decomposition of continuous sub-martingales. Theory Probab. Appl. 10 401-410. MR0202179
8. DELBAEN, F. and SCHACHERMAYER, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5 926-945. MR1384360
9. DELBAEN, F. and SCHACHERMAYER, W. (1995). The no-arbitrage property under a change of numéraire. Stochastics Stochastics Rep. 53 213-226. MR1381678
10. DUBINS, L. E. and SCHWARZ, G. (1965). On continuous martingales. Proc. Nat. Acad. Sci. U.S.A. 53 913-916. MR0178499
11. DUFFIE, D. (1996). Dynamic Asset Pricing Theory, 2nd ed. Princeton Univ. Press.
12. EFRON, B. and HINKLEY, D. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information (with discussion). Biometrika 65 457-487. MR0521817
13. ELERIAN, O., SIDDHARTHA, C. and SHEPHARD, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959-993. MR 1839375
14. FAN, J. (2005). A selective overview of nonparametric methods in financial econometrics (with discussion). Statist. Sci. 20 317-357. MR2210224
15. FLORENS-ZMIROU, D. (1993). On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 790-804. MR1242012
16. FÖLLMER, H. and SONDERMANN, D. (1986). Hedging of non-redundant contingent claims. In Contributions to Mathematical Economics (W. Hildenbrand and A. Mas-Colell, eds.) 205-223. North Holland, Amsterdam.
17. FOSTER, D. and NELSON, D. B. (1996). Continuous record asymptotics for rolling sample variance estimators. Econometrica 64 139-174. MR1366144
18. GENON-CATALOT, V. and JACOD, J. (1994). Estimation of the diffusion coefficient for diffusion processes: Random sampling. Scand. J. Statist. 21 193-221. MR1292636
19. GENON-CATALOT, V., JEANTHEAU, T. and LAREDO, C. (1999). Parameter estimation for discretely observed stochastic volatility models. Bernoulli 5 855-872. MR1715442
20. GENON-CATALOT, V., JEANTHEAU, T. and LAREDO, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 1051-1079. MR1809735
21. GENON-CATALOT, V., LAREDO, C. and PICARD, D. (1992). Nonparametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Statist. 19 317-335. MR1211787
22. GLOTER, A. (2000). Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient. ESAIM Probab. Statist. 4 205-227. MR1808927
23. GLOTER, A. and JACOD, J. (2001). Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Statist. 5 225-242. MR1875672
24. HARRISON, J. and KREPS, D. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20 381-408. MR0540823
25. HARRISON, J. and PLISKA, S. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215-260. MR0622165
26. p estimation of the diffusion coefficient. Bernoulli 5 447-481. MR1693608
27. HOFFMANN, M. (2002). Rate of convergence for parametric estimation in a stochastic volatility model. Stochastic Process. Appl. 97 147-170. MR1870964
28. HULL, J. (1999). Options, Futures and Other Derivatives, 4th ed. Prentice Hall, Upper Saddle River, NJ.
29. JACOBSEN, M. (2001). Discretely observed diffusions: Classes of estimating functions and small δ-optimality. Scand. J. Statist. 28 123-149. MR1844353
30. JACOD, J. (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Math. 714. Springer, Berlin. MR0542115
31. JACOD, J. (2000). Non-parametric kernel estimation of the coefficient of a diffusion. Scand. J. Statist. 27 83-96. MR1774045
32. JACOD, J. and PROTTER, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267-307. MR1617049
33. JACOD, J. and SHIRYAEV, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, Berlin. MR1943877
34. KARATZAS, I. and SHREVE, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. MR1121940
35. MYKLAND, P. A. (1993). Asymptotic expansions for martingales. Ann. Probab. 21 800-818. MR1217566
36. MYKLAND, P. A. (1995). Dual likelihood. Ann. Statist. 23 396-421. MR1332573
37. MYKLAND, P. A. (1995). Embedding and asymptotic expansions for martingales. Probab. Theory Related Fields 103 475-492. MR1360201
38. PROTTER, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Springer, Berlin. MR2020294
39. RÉNYI, A. (1963). On stable sequences of events. Sankhyā Ser. A 25 293-302. MR0170385
40. ROOTZÉN, H. (1980). Limit distributions for the error in approximation of stochastic integrals. Ann. Probab. 2 241-251. MR0566591
41. SCHWEIZER, M. (1990). Risk-minimality and orthogonality of martingales. Stochastics Stochastics Rep. 30 123-131. MR1054195
42. SØRENSEN, H. (2001). Discretely observed diffusions: Approximation of the continuous-time score function. Scand. J. Statist. 28 113-121. MR1844352
43. VASICEK, O. (1977). An equilibrium characterization of the term structure. J. Financial Economics 5 177-188.
44. ZHANG, L. (2001). From martingales to ANOVA: Implied and realized volatility. Ph.D. dissertation, Dept. Statistics, Univ. Chicago.
45. ZHANG, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli. To appear.
46. ZHANG, L., MYKLAND, P. A. and AÏT-SAHALIA, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100 1394-1411.