# American Institute of Mathematical Sciences

March  2022, 7(1): 31-44. doi: 10.3934/puqr.2022003

## Threshold reweighted Nadaraya–Watson estimation of jump-diffusion models

 1 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, Shandong, China 2 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam 999077, Hong Kong, China 3 School of Finance and Business, Shanghai Normal University, Shanghai 200234, China

wanghanchao@sdu.edu.cn

Received  February 15, 2022 Accepted  March 11, 2022 Published  March 2022 Early access  March 2022

Fund Project: The research of Kunyang Song and Hanchao Wang is supported by the National Natural Science Foundation of China (Grant Nos.12071257 and 11971267), National Key R&D Program of China (Grant No. 2018YFA0703900), Shandong Provincial Natural Science Foundation (Grant No. ZR2019ZD41), and the Young Scholars Program of Shandong University. Yuping Song’s research is supported by the National Natural Science Foundation of China (Grant No. 11901397), Ministry of Education, Humanities and Social Sciences project (Grant No. 18YJCZH153), National Statistical Science Research Project (Grant No. 2018LZ05), Youth Academic Backbone Cultivation Project of Shanghai Normal University (Grant No. 310-AC7031-19-003021), General Research Fund of Shanghai Normal University (Grant No. SK201720), Key Subject of Quantitative Economics (Grant No. 310-AC7031-19-004221), and Academic Innovation Team of Shanghai Normal University (Grant No. 310-AC7031-19-004228).

In this paper, we propose a new method to estimate the diffusion function in the jump-diffusion model. First, a threshold reweighted Nadaraya–Watson-type estimator is introduced. Then, we establish asymptotic normality for the estimator and conduct Monte Carlo simulations through two examples to verify the better finite-sampling properties. Finally, our estimator is demonstrated through the actual data of the Shanghai Interbank Offered Rate in China.

Citation: Kunyang Song, Yuping Song, Hanchao Wang. Threshold reweighted Nadaraya–Watson estimation of jump-diffusion models. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 31-44. doi: 10.3934/puqr.2022003
##### References:
 [1] Aït-Sahalia, Y. and Jacod, J., High-Frequency Financial Economet rics, Princeton University Press, 2014. [2] Bandi, F. M. and Nguyen, T. H., On the functional estimation of jump-diffusion models, J. Econometrics, 2003, 116(1/2): 293−328. [3] Bandi, F. M. and Phillips, P. C. B., Fully nonparametric estimation of scalar diffusion models, Econometrica, 2003, 71(1): 241−283. doi: 10.1111/1468-0262.00395. [4] Barndorff-Nielsen, O. E. and Shephard, N., Econometrics of testing for jumps in financial economics using bipower variation, J. Financial Econometrics, 2006, 4(1): 1−30. [5] Fan, J., Fan, Y. and Jiang, J., Dynamic integration of time- and state-domain methods for volatility estimation, Journal of American Statistical Association, 2007, 102(478): 618−631. doi: 10.1198/016214507000000176. [6] Florens-Zmirou, D., On estimating the diffusion coefficient from discrete observations, J. Appl. Probab., 1993, 30(4): 790−804. doi: 10.2307/3214513. [7] Hanif, M., Wang, H. and Lin, Z., Reweighted Nadaraya-Watson estimation of jump-diffusion models, Science China Mathematics, 2012, 55(5): 1005−1016. doi: 10.1007/s11425-011-4340-4. [8] Jacod, J., Statistics and high-frequency data, Statistical Methods for Stochastic Differential Equations, 2012, 191–310, MR2976984. [9] Jacod, J. and Shiryaev, A., Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften Springer, 2003. [10] Jiang, G. and Knight, J., A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model, Econometric Theory, 1997, 13(5): 615−645. [11] Lin, Z. and Wang, H., Empirical likelihood inference for diffusion processes with jumps, Science China Mathematics, 2010, 53(7): 1805−1816. doi: 10.1007/s11425-010-4027-2. [12] Mancini, C., Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps, Scand. J. Stat., 2009, 36(2): 270−296. doi: 10.1111/j.1467-9469.2008.00622.x. [13] Mancini, C. and Renò, R., Threshold estimation of Markov models with jumps and interest rate modeling, J. Econometrics, 2011, 160(1): 77−92. doi: 10.1016/j.jeconom.2010.03.019. [14] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften, Springer, 1999. [15] Rogers, L. and Williams, D., Diffusions, Markov Processes and Martingales, Volume 2: Itô Calculus, Cambridge University Press, 2000. [16] Situ, R., Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer Science & Business Media, 2005. [17] Song, Y. and Wang, H., Central limit theorems of local polynomial threshold estimator for diffusion processes with jumps, Scand. J. Stat., 2018, 45(3): 644−681. doi: 10.1111/sjos.12318. [18] Xu, K. L., Empirical likelihood based inference for nonparametric recurrent diffusions, J. Econometrics., 2009, 153(1): 65−82. doi: 10.1016/j.jeconom.2009.04.006. [19] Xu, K. L., Re-weighted functional estimation of diffusion models, Econometric Theory, 2010, 26(2): 541−563. doi: 10.1017/S0266466609100087.

show all references

##### References:
 [1] Aït-Sahalia, Y. and Jacod, J., High-Frequency Financial Economet rics, Princeton University Press, 2014. [2] Bandi, F. M. and Nguyen, T. H., On the functional estimation of jump-diffusion models, J. Econometrics, 2003, 116(1/2): 293−328. [3] Bandi, F. M. and Phillips, P. C. B., Fully nonparametric estimation of scalar diffusion models, Econometrica, 2003, 71(1): 241−283. doi: 10.1111/1468-0262.00395. [4] Barndorff-Nielsen, O. E. and Shephard, N., Econometrics of testing for jumps in financial economics using bipower variation, J. Financial Econometrics, 2006, 4(1): 1−30. [5] Fan, J., Fan, Y. and Jiang, J., Dynamic integration of time- and state-domain methods for volatility estimation, Journal of American Statistical Association, 2007, 102(478): 618−631. doi: 10.1198/016214507000000176. [6] Florens-Zmirou, D., On estimating the diffusion coefficient from discrete observations, J. Appl. Probab., 1993, 30(4): 790−804. doi: 10.2307/3214513. [7] Hanif, M., Wang, H. and Lin, Z., Reweighted Nadaraya-Watson estimation of jump-diffusion models, Science China Mathematics, 2012, 55(5): 1005−1016. doi: 10.1007/s11425-011-4340-4. [8] Jacod, J., Statistics and high-frequency data, Statistical Methods for Stochastic Differential Equations, 2012, 191–310, MR2976984. [9] Jacod, J. and Shiryaev, A., Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften Springer, 2003. [10] Jiang, G. and Knight, J., A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model, Econometric Theory, 1997, 13(5): 615−645. [11] Lin, Z. and Wang, H., Empirical likelihood inference for diffusion processes with jumps, Science China Mathematics, 2010, 53(7): 1805−1816. doi: 10.1007/s11425-010-4027-2. [12] Mancini, C., Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps, Scand. J. Stat., 2009, 36(2): 270−296. doi: 10.1111/j.1467-9469.2008.00622.x. [13] Mancini, C. and Renò, R., Threshold estimation of Markov models with jumps and interest rate modeling, J. Econometrics, 2011, 160(1): 77−92. doi: 10.1016/j.jeconom.2010.03.019. [14] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften, Springer, 1999. [15] Rogers, L. and Williams, D., Diffusions, Markov Processes and Martingales, Volume 2: Itô Calculus, Cambridge University Press, 2000. [16] Situ, R., Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer Science & Business Media, 2005. [17] Song, Y. and Wang, H., Central limit theorems of local polynomial threshold estimator for diffusion processes with jumps, Scand. J. Stat., 2018, 45(3): 644−681. doi: 10.1111/sjos.12318. [18] Xu, K. L., Empirical likelihood based inference for nonparametric recurrent diffusions, J. Econometrics., 2009, 153(1): 65−82. doi: 10.1016/j.jeconom.2009.04.006. [19] Xu, K. L., Re-weighted functional estimation of diffusion models, Econometric Theory, 2010, 26(2): 541−563. doi: 10.1017/S0266466609100087.
One path of process $X_t$ in model (3.1)
RNW and NW estimators for values of $x$ of the path $X_t$ in model (3.1)
The QQ plot for $\sigma^2(x)$ in model (3.1)
One path of process $X_t$ in model (3.2)
RNW and NW estimators for different point $x$ of the path $X_t$ in model (3.2)
The QQ plot for $\sigma^2(x)$ in model (3.2)
Time Series of the one-month Shibor
Values of various measures and estimators under different time spans and sampling numbers for the diffusion function in model (3.1)
 Measures Estimators $T = 5$ $T = 10$ $T = 20$ $n = 500$ MSE NW 1.32E-08 6.91E-09 5.20E-08 RNW 6.50E-09 6.40E-09 5.11E-08 RMSE NW 1.15E-04 8.31E-05 2.28E-04 RNW 8.06E-05 8.00E-05 2.26E-04 MADE NW 8.98E-05 7.91E-05 2.20E-04 RNW 7.13E-05 7.51E-05 2.17E-04 $n = 1000$ MSE NW 2.02E-09 1.19E-09 1.15E-08 RNW 1.54E-09 9.32E-10 1.11E-08 RMSE NW 4.50E-05 3.44E-05 1.07E-04 RNW 3.92E-05 3.05E-05 1.05E-04 MADE NW 3.03E-05 2.56E-05 1.05E-04 RNW 3.21E-05 2.18E-05 1.03E-04 $n = 2000$ MSE NW 1.35E-09 3.84E-09 7.23E-09 RNW 1.19E-09 3.80E-09 7.15E-09 RMSE NW 3.67E-05 6.19E-05 8.51E-05 RNW 3.45E-05 6.16E-05 8.45E-05 MADE NW 3.27E-05 5.55E-05 8.11E-05 RNW 2.69E-05 5.59E-05 8.07E-05 1 Note: 1.32E-08 denotes 1.32×10−8
 Measures Estimators $T = 5$ $T = 10$ $T = 20$ $n = 500$ MSE NW 1.32E-08 6.91E-09 5.20E-08 RNW 6.50E-09 6.40E-09 5.11E-08 RMSE NW 1.15E-04 8.31E-05 2.28E-04 RNW 8.06E-05 8.00E-05 2.26E-04 MADE NW 8.98E-05 7.91E-05 2.20E-04 RNW 7.13E-05 7.51E-05 2.17E-04 $n = 1000$ MSE NW 2.02E-09 1.19E-09 1.15E-08 RNW 1.54E-09 9.32E-10 1.11E-08 RMSE NW 4.50E-05 3.44E-05 1.07E-04 RNW 3.92E-05 3.05E-05 1.05E-04 MADE NW 3.03E-05 2.56E-05 1.05E-04 RNW 3.21E-05 2.18E-05 1.03E-04 $n = 2000$ MSE NW 1.35E-09 3.84E-09 7.23E-09 RNW 1.19E-09 3.80E-09 7.15E-09 RMSE NW 3.67E-05 6.19E-05 8.51E-05 RNW 3.45E-05 6.16E-05 8.45E-05 MADE NW 3.27E-05 5.55E-05 8.11E-05 RNW 2.69E-05 5.59E-05 8.07E-05 1 Note: 1.32E-08 denotes 1.32×10−8
Values of various measures and estimators under different time spans and sampling numbers for the diffusion function in model (3.2)
 Measures Estimators $T = 5$ $T = 10$ $T = 20$ $n = 500$ MSE NW 3.58E-05 2.03E-05 1.86E-02 RNW 2.31E-05 1.73E-05 7.69E-03 RMSE NW 5.98E-03 4.50E-03 1.36E-01 RNW 4.81E-03 4.17E-03 8.77E-02 MADE NW 5.79E-03 3.77E-03 9.82E-02 RNW 4.70E-03 3.57E-03 6.06E-02 $n = 1000$ MSE NW 6.62E-06 6.75E-05 8.89E-04 RNW 5.21E-06 2.82E-05 6.86E-04 RMSE NW 2.57E-03 8.22E-03 2.98E-02 RNW 2.28E-03 5.31E-03 2.62E-02 MADE NW 2.19E-03 6.27E-03 2.59E-02 RNW 1.87E-03 4.10E-03 2.04E-02 $n = 2000$ MSE NW 1.40E-06 1.38E-04 1.31E-05 RNW 1.10E-06 4.57E-05 1.81E-06 RMSE NW 1.18E-03 1.18E-02 3.62E-03 RNW 1.05E-03 6.76E-03 1.35E-03 MADE NW 8.52E-04 1.02E-02 3.01E-03 RNW 7.56E-04 4.34E-03 1.11E-03 1 Note: 3.58E-05 denotes 3.58×10−5.
 Measures Estimators $T = 5$ $T = 10$ $T = 20$ $n = 500$ MSE NW 3.58E-05 2.03E-05 1.86E-02 RNW 2.31E-05 1.73E-05 7.69E-03 RMSE NW 5.98E-03 4.50E-03 1.36E-01 RNW 4.81E-03 4.17E-03 8.77E-02 MADE NW 5.79E-03 3.77E-03 9.82E-02 RNW 4.70E-03 3.57E-03 6.06E-02 $n = 1000$ MSE NW 6.62E-06 6.75E-05 8.89E-04 RNW 5.21E-06 2.82E-05 6.86E-04 RMSE NW 2.57E-03 8.22E-03 2.98E-02 RNW 2.28E-03 5.31E-03 2.62E-02 MADE NW 2.19E-03 6.27E-03 2.59E-02 RNW 1.87E-03 4.10E-03 2.04E-02 $n = 2000$ MSE NW 1.40E-06 1.38E-04 1.31E-05 RNW 1.10E-06 4.57E-05 1.81E-06 RMSE NW 1.18E-03 1.18E-02 3.62E-03 RNW 1.05E-03 6.76E-03 1.35E-03 MADE NW 8.52E-04 1.02E-02 3.01E-03 RNW 7.56E-04 4.34E-03 1.11E-03 1 Note: 3.58E-05 denotes 3.58×10−5.
Comparisons between NW and RNW estimators for $\sigma^{2}(x)$ with various measures and different bandwidths
 Measure $h = c \cdot \hat{S} \cdot n^{-\frac{1}{5}}$ $c = 1.06$ $c = 3$ $c = 5$ NW RNW NW RNW NW RNW MSE 0.999917 0.999910 0.999902 0.999900 0.999909 0.999900 RMSE 0.999959 0.999955 0.999951 0.999950 0.999954 0.999950 MADE 0.999959 0.999955 0.999951 0.999950 0.999954 0.999950 RADE 0.999979 0.999977 0.999976 0.999975 0.999977 0.999975
 Measure $h = c \cdot \hat{S} \cdot n^{-\frac{1}{5}}$ $c = 1.06$ $c = 3$ $c = 5$ NW RNW NW RNW NW RNW MSE 0.999917 0.999910 0.999902 0.999900 0.999909 0.999900 RMSE 0.999959 0.999955 0.999951 0.999950 0.999954 0.999950 MADE 0.999959 0.999955 0.999951 0.999950 0.999954 0.999950 RADE 0.999979 0.999977 0.999976 0.999975 0.999977 0.999975
 [1] Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial and Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044 [2] Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072 [3] Chuancun Yin, Kam Chuen Yuen. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1247-1262. doi: 10.3934/jimo.2015.11.1247 [4] Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057 [5] Xinhong Zhang, Qing Yang. Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3155-3175. doi: 10.3934/dcdsb.2021177 [6] Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100 [7] Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003 [8] Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048 [9] Johnathan M. Bardsley. A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Problems and Imaging, 2010, 4 (1) : 11-17. doi: 10.3934/ipi.2010.4.11 [10] Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control and Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21 [11] Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092 [12] Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control and Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025 [13] Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298 [14] Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069 [15] Caibin Zhang, Zhibin Liang, Kam Chuen Yuen. Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach. Journal of Industrial and Management Optimization, 2022, 18 (1) : 341-366. doi: 10.3934/jimo.2020156 [16] Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026 [17] Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022068 [18] Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems and Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167 [19] Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385 [20] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191

Impact Factor: