Binomial jump-amplitude modeling in SDEs: A regularized stepwise estimation with computational diagnostics

Wuchen Li; Zhaoxiang Xu; Jian Xu; Linghui Li; Liping Bai; Wuchen Li; Zhaoxiang Xu; Jian Xu; Linghui Li; Liping Bai

doi:10.3934/math.20251186

AIMS Mathematics

2025, Volume 10, Issue 11: 26994-27015. doi: 10.3934/math.20251186

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Binomial jump-amplitude modeling in SDEs: A regularized stepwise estimation with computational diagnostics

1.
Faculty of Innovation Engineering, Macau University of Science and Technology, Taipa, Macau, China
2.
School of Fashion and Textiles, The Hong Kong Polytechnic University, Hong Kong, China
3.
South China University of Technology, Guangzhou, China

Received: 07 August 2025 Revised: 30 October 2025 Accepted: 05 November 2025 Published: 20 November 2025
MSC : 60H35

The presence of the jump process makes parameter estimation for stationary stochastic differential equations particularly challenging. Moreover, existing jump parameter models often suffer from significant systematic errors. This paper introduces a new stationary stochastic differential equation model with jumps, in which the jump amplitude follows a binomial distribution. This approach helps mitigate systematic errors, particularly those arising when the probability density remains nonzero for infinitely large jump amplitudes or when it becomes excessively high at zero jump size. On this basis, we use the stepwise estimation method to estimate the parameters of the model (that is, first estimate parameters of the drift and diffusion term by the tool of quadratic variation, and then estimate the parameters of the jump process), and the result has a high estimation accuracy.
- stochastic differential equations,
- jump amplitude,
- quadratic variation,
- binomial distribution,
- stepwise parameter estimation
Citation: Wuchen Li, Zhaoxiang Xu, Jian Xu, Linghui Li, Liping Bai. Binomial jump-amplitude modeling in SDEs: A regularized stepwise estimation with computational diagnostics[J]. AIMS Mathematics, 2025, 10(11): 26994-27015. doi: 10.3934/math.20251186

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Abstract

The presence of the jump process makes parameter estimation for stationary stochastic differential equations particularly challenging. Moreover, existing jump parameter models often suffer from significant systematic errors. This paper introduces a new stationary stochastic differential equation model with jumps, in which the jump amplitude follows a binomial distribution. This approach helps mitigate systematic errors, particularly those arising when the probability density remains nonzero for infinitely large jump amplitudes or when it becomes excessively high at zero jump size. On this basis, we use the stepwise estimation method to estimate the parameters of the model (that is, first estimate parameters of the drift and diffusion term by the tool of quadratic variation, and then estimate the parameters of the jump process), and the result has a high estimation accuracy.

References

[1]	I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, New York: Springer-Verlag, 1991.
[2]	I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, New York: Springer, 1998. https://doi.org/10.1007/978-1-4612-0949-2
[3]	B. Øksendal, Stochastic differential equations: An introduction with applications, Springer, 2003.
[4]	D. W. Stroock, S. S. Varadhan, Multidimensional diffusion processes, Springer, 1979.
[5]	R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3 (1976), 125–144. https://doi.org/10.1016/0304-405X(76)90022-2 doi: 10.1016/0304-405X(76)90022-2
[6]	P. Carr, L. Wu, The finite moment log stable process and option pricing, J. Finance, 58 (2003), 753–777. https://doi.org/10.1111/1540-6261.00544 doi: 10.1111/1540-6261.00544
[7]	A. Dassios, J. Jang, Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity, Finance Stochast., 7 (2003), 73–95. https://doi.org/10.1007/s007800200079 doi: 10.1007/s007800200079
[8]	F. B. Hanson, J. J. Westman, Jump-diffusion stock return models in finance: Stochastic process density with uniform-jump amplitude, In: 15th International Symposium on Mathematical Theory of Networks and Systems, 2002.
[9]	C. A. Ramezani, Y. Zeng, Maximum likelihood estimation of the double exponential jump-diffusion process, Ann. Finance, 3 (2007), 487–507. https://doi.org/10.1007/s10436-006-0062-y doi: 10.1007/s10436-006-0062-y
[10]	S. G. Kou, A jump-diffusion model for option pricing, Manage. Sci., 48 (2002), 1086–1101.
[11]	R. Cont, P. Tankov, Financial modelling with jump processes, New York: Chapman and Hall/CRC, 2004. https://doi.org/10.1201/9780203485217
[12]	Y. Aït-Sahalia, J. Jacod, Estimating the degree of activity of jumps in high frequency data, Ann. Statist., 37 (2009), 2202–2244. https://doi.org/10.1214/08-AOS640 doi: 10.1214/08-AOS640
[13]	E. F. Fama, R. Roll, Some properties of symmetric stable distributions, J. Amer. Statist. Assoc., 63 (1968), 817–836. https://doi.org/10.1080/01621459.1968.11009311 doi: 10.1080/01621459.1968.11009311
[14]	X. Gabaix, Power laws in economics and finance, Annu. Rev. Econ., 1 (2009), 255–294. 10.1146/annurev.economics.050708.142940 doi: 10.1146/annurev.economics.050708.142940
[15]	D. B. Madan, E. Seneta, The variance gamma (VG) model for share market returns, J. Bus., 63 (1990), 511–524. https://doi.org/10.1086/296519 doi: 10.1086/296519
[16]	T. G. Andersen, L. Benzoni, J. Lund, An empirical investigation of continuous-time equity return models, J. Finance, 57 (2002), 1239–1284. https://doi.org/10.1111/1540-6261.00460 doi: 10.1111/1540-6261.00460
[17]	D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Rev. Financ. Stud., 9 (1996), 69–107. https://doi.org/10.1093/rfs/9.1.69 doi: 10.1093/rfs/9.1.69
[18]	D. Applebaum, Lévy processes and stochastic calculus, 2Eds, New York: Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511809781
[19]	J. P. Nolan, Stable distributions: Models for heavy-tailed data, Birkhäuser, 2001.
[20]	H. Rinne, The Weibull distribution: A handbook, New York: Chapman and Hall/CRC, 2008. https://doi.org/10.1201/9781420087444
[21]	Z. A. Karian, E. J. Dudewicz, Handbook of fitting statistical distributions with R, CRC Press, 2010.
[22]	N. L. Johnson, A. W. Kemp, S. Kotz, Univariate discrete distributions, John Wiley & Sons, 2005. https://doi.org/10.1002/0471715816
[23]	J. Li, V. Todorov, G. Tauchen, Volatility occupation times, Ann. Statist., 41 (2013), 1865–1891. https://doi.org/10.1214/13-AOS1135
[24]	J. Li, V. Todorov, G. Tauchen, Estimating the volatility occupation time via regularized Laplace inversion, Economet. Theor., 32 (2016), 1253–1288. https://doi.org/10.1017/S0266466615000171 doi: 10.1017/S0266466615000171
[25]	S. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer, 2004.
[26]	O. E. Barndorff-Nielsen, N. Shephard, Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. B, 63 (2001), 167–241. https://doi.org/10.1111/1467-9868.00282 doi: 10.1111/1467-9868.00282
[27]	O. Vasicek, An equilibrium characterization of the term structure, J. Financ. Econ., 5 (1977), 177–188. https://doi.org/10.1016/0304-405X(77)90016-2 doi: 10.1016/0304-405X(77)90016-2
[28]	D. T. Gillespie, Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral, Phys. Rev. E, 54 (1996), 2084–2091. https://doi.org/10.1103/PhysRevE.54.2084 doi: 10.1103/PhysRevE.54.2084
[29]	A. R. Pedersen, A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scand. J. Stat., 22 (1995), 55–71.
[30]	C. Fuchs, Inference for diffusion processes: With applications in life sciences, Springer, 2013.
[31]	P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Berlin, Heidelberg: Springer, 1992. https://doi.org/10.1007/978-3-662-12616-5
[32]	Y. Aït-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach, Econometrica, 70 (2002), 223–262. https://doi.org/10.1111/1468-0262.00274 doi: 10.1111/1468-0262.00274
[33]	T. Hastie, R. Tibshirani, J. H. Friedman, The elements of statistical learning, Springer Science & Business Media, 2001.
[34]	H. Zou, T. Hastie, Regularization and variable selection via the elastic net, J. R. Stat. Soc. B, 67 (2005), 301–320. https://doi.org/10.1111/j.1467-9868.2005.00503.x doi: 10.1111/j.1467-9868.2005.00503.x
[35]	R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. B, 58 (1996), 267–288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x doi: 10.1111/j.2517-6161.1996.tb02080.x
[36]	S. V. Lototsky, B. L. Rozovskii, Stochastic partial differential equations, Springer, 2015.
[37]	P. Craigmile, R. Herbei, G. Liu, G. Schneider, Statistical inference for stochastic differential equations, WIREs Comput. Stat., 15 (2022), e1585. https://doi.org/10.1002/wics.1585 doi: 10.1002/wics.1585
[38]	G. B. Durham, A. R. Gallant, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, J. Bus. Econom. Statist., 20 (2002), 297–316.
[39]	W. Li, Y. Jiang, J. Xu, L. Wang, L. Bai, Enhanced parameter estimation for stable SDEs with jumps via quadratic variation, J. Stat. Comput. Sim., 95 (2025), 2932–2948. https://doi.org/10.1080/00949655.2025.2511924 doi: 10.1080/00949655.2025.2511924
[40]	W. Li, L. Zhang, J. Xu, L. Li, L. Bai, Jump amplitude inference in SDEs with cosine kernel, Results Appl. Math., 26 (2025), 100596. https://doi.org/10.1016/j.rinam.2025.100596 doi: 10.1016/j.rinam.2025.100596
[41]	R. Hu, D. Zhang, X. Gu, Reliability analysis of a class of stochastically excited nonlinear Markovian jump systems, Chaos Soliton. Fract., 155 (2022), 111737. https://doi.org/10.1016/j.chaos.2021.111737 doi: 10.1016/j.chaos.2021.111737
[42]	A. Delaigle, I. Gijbels, Practical bandwidth selection in deconvolution kernel density estimation, Comput. Stat. Data An., 45 (2004), 249–267. https://doi.org/10.1016/S0167-9473(02)00329-8 doi: 10.1016/S0167-9473(02)00329-8
[43]	M. C. Jones, J. S. Marron, S. J. Sheather, A brief survey of bandwidth selection for density estimation, J. Am. Stat. Assoc., 91 (1996), 401–407.
[44]	B. W. Silverman, Density estimation for statistics and data analysis, Chapman and Hall/CRC, 1986.
[45]	X. D. Gu, Y. Y. Zhang, I. Mughal, Z. C. Deng, Stochastic responses of nonlinear inclined cables with an attached damper and random excitations, Nonlinear Dynam., 112 (2024), 15969–15986. https://doi.org/10.1007/s11071-024-09877-1 doi: 10.1007/s11071-024-09877-1

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