### Nonparametric Two-Step Estimation of Drift Function in the Jump-Diffusion Model with Noisy Data

YE Xuguo1, ZHAO Yanyong2, LIN Jinguan2, LONG Weifang1

1. 1. School of Science, Kaili University, Kaili 556011, China;
2. Department of Statistics, Nanjing Audit University, Nanjing 211815, China
• Received:2021-02-24 Revised:2021-07-06 Online:2022-11-25 Published:2022-12-23
• Contact: ZHAO Yanyong,Email:zhaoyanyong1987@163.com
• Supported by:
Ye's research is supported by the National Natural Science Foundation of China under Grant No. 11961038, Young Talents Project of Science and Technology Research Program of Education Department in Guizhou Province (Qianjiao KYword[2018]364), Science and Technology Foundation of Guizhou Province (QianKeHejichu[2019]1286), Cultivating Project of National Natural Science Foundation (QianKeHe talent-development platform[2017]No. 5723, QianKeHe talent-development platform[2017]No. 5723-02). Zhao's research is supported by the National Natural Science Foundation of China under Grant Nos. 12071220, 11701286, Social Science Foundation of Jiangsu Province under Grant No. 20EYC008, the National Statistical Research Project of China under Grant No. 2020LZ35, and Open Project of Jiangsu Key Laboratory of Financial Engineering under Grant No. NSK2021-12. Lin's research is supported by the National Natural Science Foundation of China under Grant Nos. 11831008, 11971235, and the National Statistical Research Project of China under Grant No. 2020LZ19.

YE Xuguo, ZHAO Yanyong, LIN Jinguan, LONG Weifang. Nonparametric Two-Step Estimation of Drift Function in the Jump-Diffusion Model with Noisy Data[J]. Journal of Systems Science and Complexity, 2022, 35(6): 2398-2429.

This paper considers a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable. A two-step approach to estimate the drift function of a jump-diffusion model {in noisy settings} is proposed. The proposed estimator is shown to be consistent and asymptotically normal in the presence of finite activity jumps. Simulated experiments and a real data application are undertaken to assess the finite sample performance of the newly proposed method.
 [1] Johannes M, The statistical and economic role of jumps in continuous-time interest rate models, Journal of Finance, 2004, 59(1): 227–260.[2] Aït-Sahalia Y and Jacod J, Testing for jumps in a discretely observed process, Annals of Statistics, 2009, 37(1): 184–222.[3] Cont R and Tankov P, Financial Modelling with Jump Processes, Chapman & Hall/CRC Press, London, 2004.[4] Mancini C and Renò R, Threshold estimation of Markov models with jumps and interest rate modeling, Journal of Econometrics, 2011, 160(1): 77–192.[5] Protter P E, Stochastic Integration and Differential Equations, Springer-Heidelberg, Berlin, 2004.[6] Liu Q, Liu Y, Liu Z, et al., Estimation of spot volatility with superposed noisy data, The North American Journal of Economics and Finance, 2018, 44: 62–79.[7] Lo A W and Wang J, Implementing option pricing models when asset returns are predictable, Journal of Finance, 1995, 50(1): 87–129.[8] Miscia O D, Nonparametric estimation of diffusion process: A closer look, Working Paper, 2004.[9] Arfi M, Nonparametric drift estimation from ergodic samples, Journal of Nonparametric Statistics, 1995, 5(4): 381–389.[10] Stanton R, A nonparametric model of term structure dynamics and the market price of interest rate risk, Journal of Finance, 1997, 52(5): 1973–2002.[11] Bandi F M and Phillips P C, Fully nonparametric estimation of scalar diffusion models, Econometrica, 2003, 71(1): 241–283.[12] Dalalyan A, Sharp adaptive estimation of the drift function for ergodic diffusions, Annals of Statistics, 2005, 33(6): 2507–2528.[13] Galtchouk L and Pergamenshchikov S, Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes, Sequential Analysis, 2005, 24(3): 303–330.[14] Comte F, Genon-Catalot V, and Rozenholc Y, Penalized nonparametric mean square estimation of the coefficients of diffusion processes, Bernoulli, 2007, 13(2): 514–543.[15] Löcherbach E, Loukianova D, and Loukianov O, Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process, ESAIM: Probability and Statistics, 2011, 15: 197–216.[16] Van Der Meulen F, Schauer M, and Van Zanten H, Reversible jump MCMC for nonparametric drift estimation for diffusion processes, Computational Statistics & Data Analysis, 2014, 71: 615–632.[17] Bandi F M and Nguyen T H, On the functional estimation of jump-diffusion models, Journal of Econometrics, 2003, 116(1): 293–328.[18] Hanif M, Wang H, and Lin Z, Reweighted Nadaraya-Watson estimation of jump-diffusion models, Science China Mathematics, 2012, 55(5): 1005–1016.[19] Hanif M, Local linear estimation of jump-diffusion models by using asymmetric kernels, Stochastic Analysis and Applications, 2013, 31(6): 956–974.[20] Schmisser E, Non-parametric adaptive estimation of the drift for a jump diffusion process, Stochastic Processes and Their Applications, 2014, 124(1): 883–914.[21] Schmisser E, Non-parametric estimation of the diffusion coefficients of a diffusion with jumps, Stochastic Processes and Their Applications, 2019, 129(12): 5364–5405.[22] Favetto B, Parameter estimation by contrast minimization for noisy observations of a diffusion process, Statistics, 2014, 48(6): 1344–1370.[23] Schmisser E, Non-parametric drift estimation for diffusions from noisy data, Statistics & Decisions, 2011, 28(2): 119–150.[24] Lee W, Greenwood P E, Heckman N, et al., Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise, Statistical Inference for Stochastic Processes, 2017, 20(2): 237–252.[25] Ye X G, Lin J G, and Zhao Y Y, A two-step estimation of diffusion processes using noisy observations, Journal of Nonparametric Statistics, 2018, 30(1): 145–181.[26] Jing B Y, Liu Z, and Kong X B, On the estimation of integrated volatility with jumps and microstructure noise, Journal of Business & Economic Statistics, 2014, 32(3): 457–467.[27] Zhou B, High-frequency data and volatility in foreign-exchange rates, Journal of Business & Economic Statistics, 1996, 14(1): 45–52.[28] Jones C S, Nonlinear mean reversion in the short-term interest rate, Review of Financial Studies, 2003, 16(3): 793–843.[29] Aït-Sahalia Y, Nonparametric pricing of interest rate derivative securities, Econometrica, 1996, 64(3): 527–560.[30] Jacod J, Li Y, Mykland P A, et al., Microstructure noise in the continuous case: The pre-averaging approach, Stochastic Processes and Their Applications, 2009, 119(7): 2249–2276.[31] Durham G B, Likelihood-based specification analysis of continuous-time models of the short-term interest rate, Journal of Financial Economics, 2003, 70(3): 463–487.[32] Bandi F M, Short-term interest rate dynamics: A spatial approach, Journal of Financial Economics, 2002, 65(1): 73–110.[33] Hong Y and Li H, Nonparametric specification testing for continuous-time models with applications to term structure of interest rates, Review of Financial Studies, 2005, 18(1): 37–84.[34] Hamilton J D, The daily market for federal funds, Journal of Political Economy, 1996, 104(1): 26–56.[35] Florens-Zmirou D, On estimating the diffusion coefficient from discrete observations, Journal of Applied Probability, 1993, 30(4): 790–804.[36] Aït-Sahalia Y and Park J Y, Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models, Journal of Econometrics, 2016, 192(1): 119–138.[37] Wang H and Zhou L, Bandwidth selection of nonparametric threshold estimator in jump-diffusion models, Computers & Mathematics with Applications, 2017, 73(2): 211–219.[38] Jacod J and Todorov V, Efficient estimation of integrated volatility in presence of infinite variation jumps, Annals of Statistics, 2014, 42(3): 1029–1069.[39] Kong X, Liu Z, and Jing B, Testing for pure-jump processes for high-frequency data, Annals of Statistics, 2015, 43(2): 847–877.[40] Mancini C, Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps, Scandinavian Journal of Statistics, 2009, 36(2): 270–296.
 No related articles found!
Viewed
Full text

Abstract