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Realized Laplace transforms for pure-jump semimartingales. (English) Zbl 1274.62191

Summary: We consider specification and inference for the stochastic scale of discretely-observed pure-jump semimartingales with locally stable Lévy densities in the setting where both the time span of the data set increases, and the mesh of the observation grid decreases. The estimation is based on constructing a nonparametric estimate for the empirical Laplace transform of the stochastic scale over a given interval of time by aggregating high-frequency increments of the observed process on that time interval into a statistic we call realized Laplace transform. The realized Laplace transform depends on the activity of the driving pure-jump martingale, and we consider both cases when the latter is known or has to be inferred from the data.

MSC:

62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J75 Jump processes (MSC2010)

References:

[1] Aït-Sahalia, Y. and Jacob, J. (2007). Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35 355-392. · Zbl 1114.62109 · doi:10.1214/009053606000001190
[2] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 2202-2244. · Zbl 1173.62060 · doi:10.1214/08-AOS640
[3] Andrews, B., Calder, M. and Davis, R. A. (2009). Maximum likelihood estimation for \(\alpha\)-stable autoregressive processes. Ann. Statist. 37 1946-1982. · Zbl 1168.62077 · doi:10.1214/08-AOS632
[4] Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59 817-858. · Zbl 0732.62052 · doi:10.2307/2938229
[5] Barndorff-Nielsen, O. E. 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. · Zbl 0983.60028 · doi:10.1111/1467-9868.00282
[6] Belomestny, D. (2010). Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 317-351. · Zbl 1181.62151 · doi:10.1214/09-AOS715
[7] Belomestny, D. (2011). Statistical inference for time-changed Lévy processes via composite characteristic function estimation. Ann. Statist. 39 2205-2242. · Zbl 1227.62062 · doi:10.1214/11-AOS901
[8] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[9] Boyarchenko, S. and Levendorskiĭ, S. (2002). Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Probab. 12 1261-1298. · Zbl 1015.60036 · doi:10.1214/aoap/1037125863
[10] Brockwell, P. J. (2001). Continuous-time ARMA processes. In Stochastic Processes : Theory and Methods. Handbook of Statist. 19 249-276. North-Holland, Amsterdam. · Zbl 1011.62088 · doi:10.1016/S0169-7161(01)19011-5
[11] Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13 345-382. · Zbl 1092.91022 · doi:10.1111/1467-9965.00020
[12] Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Probab. 13 984-1053. · Zbl 1048.60059 · doi:10.1214/aoap/1060202833
[13] Feuerverger, A. and Mureika, R. A. (1977). The empirical characteristic function and its applications. Ann. Statist. 5 88-97. · Zbl 0364.62051 · doi:10.1214/aos/1176343742
[14] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathemtatics 714 . Springer, Berlin. · Zbl 0414.60053 · doi:10.1007/BFb0064907
[15] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67 . Springer, Heidelberg. · Zbl 1259.60004
[16] 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. · Zbl 1018.60002
[17] Klüppelberg, C., Lindner, A. and Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. J. Appl. Probab. 41 601-622. · Zbl 1068.62093 · doi:10.1239/jap/1091543413
[18] Klüppelberg, C., Meyer-Brandis, T. and Schmidt, A. (2010). Electricity spot price modelling with a view towards extreme spike risk. Quant. Finance 10 963-974. · Zbl 1210.91155 · doi:10.1080/14697680903150496
[19] Koutrouvelis, I. A. (1980). Regression-type estimation of the parameters of stable laws. J. Amer. Statist. Assoc. 75 918-928. · Zbl 0449.62026 · doi:10.2307/2287182
[20] Kryzhniy, V. V. (2003). Regularized inversion of integral transformations of Mellin convolution type. Inverse Problems 19 1227-1240. · Zbl 1048.65128 · doi:10.1088/0266-5611/19/5/313
[21] Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12 23-68. · Zbl 1021.60076 · doi:10.1214/aoap/1015961155
[22] Neumann, M. H. and Reiß, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 223-248. · Zbl 1200.62095 · doi:10.3150/08-BEJ148
[23] Paulson, A. S., Holcomb, E. W. and Leitch, R. A. (1975). The estimation of the parameters of the stable laws. Biometrika 62 163-170. · Zbl 0309.62017 · doi:10.1093/biomet/62.1.163
[24] Rosiński, J. (2007). Tempering stable processes. Stochastic Process. Appl. 117 677-707. · Zbl 1118.60037 · doi:10.1016/j.spa.2006.10.003
[25] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press, Cambridge, UK. · Zbl 0973.60001
[26] Shimizu, Y. and Yoshida, N. (2006). Estimation of parameters for diffusion processes with jumps from discrete observations. Stat. Inference Stoch. Process. 9 227-277. · Zbl 1125.62089 · doi:10.1007/s11203-005-8114-x
[27] Todorov, V. and Tauchen, G. (2011). Limit theorems for power variations of pure-jump processes with application to activity estimation. Ann. Appl. Probab. 21 546-588. · Zbl 1215.62088 · doi:10.1214/10-AAP700
[28] Todorov, V. and Tauchen, G. (2011). Volatility jumps. J. Bus. Econom. Statist. 29 356-371. · Zbl 1219.91156 · doi:10.1198/jbes.2010.08342
[29] Todorov, V. and Tauchen, G. (2012). The realized Laplace transform of volatility. Econometrica 80 1105-1127. · Zbl 1274.91344
[30] Todorov, V., Tauchen, G. and Grynkiv, I. (2011). Realized Laplace transforms for estimation of jump diffusive volatility models. J. Econometrics 164 367-381. · Zbl 1441.62889 · doi:10.1016/j.jeconom.2011.06.016
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