Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations. (English) Zbl 1467.62162
Summary: In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process together with the estimation for positive and negative polynomial moments of the CIR process. In this study, we require the same conditions of high frequency and infinite horizon as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier in the literature. However, in the non-ergodic cases, additional assumptions on the decreasing rate are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 62F10 | Point estimation |
Keywords:
Cox-Ingersoll-Ross process; high-frequency observation; local asymptotic (mixed) normality property; local asymptotic quadraticity property; Malliavin calculus; parametric estimation; square root coefficientReferences:
| [1] | Aı̈t‐Sahalia, Y., & Jacod, J. (2007). Volatility estimators for discretely sampled Lévy processes. The Annals of Statistics, 35(1), 355-392. · Zbl 1114.62109 |
| [2] | Alfonsi, A. (2015). Affine diffusions and related processes: Simulation, theory and applications (Vol. 6). Milan: Bocconi University & Springer. · Zbl 1387.60002 |
| [3] | Alòs, E., & Ewald, C.‐O. (2008). Malliavin differentiability of the Heston volatility and applications to option pricing. Advances in Applied Probability, 40, 144-162. · Zbl 1137.91422 |
| [4] | Altmayer, M., & Neuenkirch, A. (2015). Multilevel Monte Carlo quadrature of discontinuous payoffs in the generalized Heston model using Malliavin integration by parts. SIAM Journal on Financial Mathematics, 6(1), 22-52. · Zbl 1338.60168 |
| [5] | Barczy, M., Ben Alaya, M., Kebaier, A., & Pap, G. (2018). Asymptotic properties of maximum likelihood estimator for the growth rate for a jump‐type CIR process based on continuous time observations. Stochastic Processes and their Applications, 128(4), 1135-1164. · Zbl 1390.60307 |
| [6] | Barczy, M., Ben Alaya, M., Kebaier, A., & Pap, G. (2019). Asymptotic behavior of maximum likelihood estimators for a jump‐type Heston model. Journal of Statistical Planning and Inference, 198, 139-164. · Zbl 1391.60130 |
| [7] | Barczy, M., & Pap, G. (2016). Asymptotic properties of maximum‐likelihood estimators for Heston models based on continuous time observations. Statistics, 50(2), 389-417. · Zbl 1344.60031 |
| [8] | Ben Alaya, M., & Kebaier, A. (2012). Parameter estimation for the square‐root diffusions: Ergodic and nonergodic cases. Stochastic Models, 28(4), 609-634. · Zbl 06117449 |
| [9] | Ben Alaya, M., & Kebaier, A. (2013). Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square‐root diffusions. Stochastic Analysis and Applications, 31, 552-573. · Zbl 1381.60111 |
| [10] | Benke, J. M., & Pap, G. (2017). Local asymptotic quadraticity of statistical experiments connected with a Heston model. Acta Scientiarum Mathematicarum, 83, 313-344. · Zbl 1399.62042 |
| [11] | Berkaoui, A., Bossy, M., & Diop, A. (2008). Euler scheme for SDEs with non‐Lipschitz diffusion coefficient: Strong convergence. ESAIM: Probability and Statistics, 12, 1-11. · Zbl 1183.65004 |
| [12] | Bossy, M., & Diop, A. (2007). An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form|x|^α,α ∈ [1/2, 1. Research Report (No. 5396). INRIA. |
| [13] | Clément, E., Delattre, S., & Gloter, A. (2014). Asymptotic lower bounds in estimating jumps. Bernoulli, 20(3), 1059-1096. · Zbl 1401.62136 |
| [14] | Clément, E., & Gloter, A. (2015). Local asymptotic mixed normality property for discretely observed stochastic differential equations driven by stable Lévy processes. Stochastic Processes and their Applications, 125, 2316-2352. · Zbl 1312.60075 |
| [15] | Cox, J. C., Ingersoll, J. E., & Ross, S. (1985). A theory of the term structure of interest rates. Econometrica, 53, 385-408. · Zbl 1274.91447 |
| [16] | Cuchiero, C., Filipović, D., Mayerhofer, E., & Teichmann, J. (2011). Affine processes on positive semidefinite matrices. The Annals of Applied Probability, 21(2), 397-463. · Zbl 1219.60068 |
| [17] | Davis, M. H. A., & Johansson, M. P. (2006). Malliavin Monte Carlo Greeks for jump diffusions. Stochastic Processes and Their Applications, 116, 101-129. · Zbl 1081.60040 |
| [18] | De Marco, S. (2011). Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root‐type diffusions. The Annals of Applied Probability, 21(4), 1282-1321. · Zbl 1246.60082 |
| [19] | Duffie, D., Filipović, D., & Schachermayer, W. (2003). Affine processes and applications in finance. The Annals of Applied Probability, 13(3), 984-1053. · Zbl 1048.60059 |
| [20] | Gobet, E. (2001). Local asymptotic mixed normality property for elliptic diffusions: A Malliavin calculus approach. Bernoulli, 7, 899-912. · Zbl 1003.60057 |
| [21] | Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 38, 711-737. · Zbl 1018.60076 |
| [22] | Höpfner, R. (2014). Asymptotics statistics. Berlin: De Gruyter. · Zbl 1306.62013 |
| [23] | Ikeda, N., & Watanabe, S. (1989). Stochastic differential equations and diffusion processes. Amsterdam, Netherlands: 2nd, North‐Holland‐Kodansha. · Zbl 0684.60040 |
| [24] | Jacod, J. (2012). Statistics and high frequency data. In M.Kessler (ed.), A.Lindner (ed.), & M.Sørensen (ed.) (Eds.), Statistical Methods for Stochastic Differential EquationsMonographs on statistics and applied probability (Vol. 124). Boca Raton, FL: Chapman & Hall/CRC Press. · Zbl 1375.62025 |
| [25] | Jacod, J., & Shiryaev, A. N. (2003). Limit theorems for stochastic processes (2nd ed.). Berlin, Germany: Springer‐Verlag. · Zbl 1018.60002 |
| [26] | Jeganathan, P. (1982). On the asymptotic theory of estimation when the limit of the log‐likelihood ratios is mixed normal. Sankhya A, 44(2), 173-212. · Zbl 0584.62042 |
| [27] | Jin, P., Mandrekar, V., Rüdiger, B., & Trabelsi, C. (2013). Positive Harris recurrence of the CIR process and its applications. Communications on Stochastic Analysis, 7(3), 409-424. |
| [28] | Kallsen, J. (2006). A didactic note on affine stochastic volatility models. In From stochastic calculus to mathematical finance (pp. 343-368). Berlin, Germany: Springer. · Zbl 1104.60024 |
| [29] | Kawai, R. (2013). Local asymptotic normality property for Ornstein‐Uhlenbeck processes with jumps under discrete sampling. Journal of Theoretical Probability, 26, 932-967. · Zbl 1281.62175 |
| [30] | Keller‐Ressel, M., & Mijatović, A. (2012). On the limit distributions of continuous‐state branching processes with immigration. Stochastic Processes and Their Applications, 122(6), 2329-2345. · Zbl 1242.60088 |
| [31] | Kohatsu‐Higa, A., Nualart, E., & Tran, N. K. (2014). LAN property for a simple Lévy process. Comptes Rendus Mathematique, 352(10), 859-864. · Zbl 1306.60048 |
| [32] | Kohatsu‐Higa, A., Nualart, E., & Tran, N. K. (2017). LAN property for an ergodic diffusion with jumps. Statistics, 51(2), 419-454. · Zbl 1369.60054 |
| [33] | Kohatsu‐Higa, A., & Yasuda, K. (2009). Estimating multidimensional density functions using the Malliavin‐Thalmaier formula. SIAM Journal on Numerical Analysis, 47(2), 1546-1575. · Zbl 1395.62082 |
| [34] | Le Cam, L. (1960). Locally asymptotically normal families of distributions. University of California Publications in Statistics, 3, 37-98. · Zbl 0104.12701 |
| [35] | Le Cam, L., & Lo Yang, G. (1990). Asymptotics in statistics: Some basic conceptsSpringer series in statistics (). New York, NY: Springer‐Verlag. · Zbl 0719.62003 |
| [36] | León, J. A., Navarro, R., & Nualart, D. (2003). An anticipating calculus approach to the utility maximization of an insider. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 13(1), 171-185. · Zbl 1060.91054 |
| [37] | Nualart, D. (2006). The Malliavin calculus and related topics (2nd ed.). New York, NY: Springer. · Zbl 1099.60003 |
| [38] | Overbeck, L. (1998). Estimation for continuous branching processes. Scandinavian Journal of Statistics, 25(1), 111-126. · Zbl 0905.62083 |
| [39] | Overbeck, L., & Rydén, T. (1997). Estimation in the Cox‐Ingersoll‐Ross model. Econometric Theory, 13(3), 430-461. |
| [40] | Pagès, G. (2015). Introduction to numerical probability for finance. Paris, France: LPMA‐Université Pierre et Marie Curie. |
| [41] | Petrou, E. (2008). Malliavin calculus in Lévy spaces and applications to finance. Electronic Journal of Probability, 13, 852-879. · Zbl 1193.60075 |
| [42] | Pinsky, M. A. (1972). Limit theorems for continuous state branching processes with immigration. Bulletin of the American Mathematical Society, 78(2), 242-244. · Zbl 0252.60031 |
| [43] | Protter, P. (2005). Stochastic integration and differential equations. Berlin Heidelberg/Germany and New York, NY: 2nd, Springer. |
| [44] | Tran, N. K. (2017). LAN property for an ergodic Ornstein‐Uhlenbeck process with Poisson jumps. Communications in Statistics‐Theory and Methods, 46(16), 7942-7968. · Zbl 1372.60077 |
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