June  2022, 4(2): 299-322. doi: 10.3934/fods.2022008

Unbiased parameter inference for a class of partially observed Lévy-process models

Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Kingdom of Saudi Arabia

* Corresponding author: Hamza Ruzayqat

Received  December 2021 Published  June 2022 Early access  April 2022

Fund Project: Both authors are supported by KAUST baseline funding

We consider the problem of static Bayesian inference for partially observed Lévy-process models. We develop a methodology which allows one to infer static parameters and some states of the process, without a bias from the time-discretization of the afore-mentioned Lévy process. The unbiased method is exceptionally amenable to parallel implementation and can be computationally efficient relative to competing approaches. We implement the method on S & P 500 log-return daily data and compare it to some Markov chain Monte Carlo (MCMC) algorithm.

Citation: Hamza Ruzayqat, Ajay Jasra. Unbiased parameter inference for a class of partially observed Lévy-process models. Foundations of Data Science, 2022, 4 (2) : 299-322. doi: 10.3934/fods.2022008
References:
[1]

C. AndrieuA. Doucet and R. Holenstein, Particle Markov chain Monte Carlo methods, J. R. Stat. Soc. Ser. B Stat. Methodol., 72 (2010), 269-342.  doi: 10.1111/j.1467-9868.2009.00736.x.

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323.

[3]

O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0197-7.

[4]

O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian OU-based models and some of their uses in financial economics, J. Roy. Statist. Soc. B Stat. Methodol., 63 (2001), 167-241.  doi: 10.1111/1467-9868.00282.

[5]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996.

[6]

R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech., 10 (1961), 493-516. 

[7]

O. Cappé, T. Ryden and E. Moulines, Inference in Hidden Markov Models, Springer, New York, 2005.

[8]

N. K. ChadaJ. FranksA. JasraK. J. H. Law and M. Vihola, Unbiased inference for discretely observed hidden markov model diffusions, SIAM/ASA J. Uncertain. Quantif., 9 (2021), 763-787.  doi: 10.1137/20M131549X.

[9]

P. Del Moral, Feyman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9393-1.

[10]

S. Dereich and F. Heidenreich, A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations, Stoc. Proc. Appl., 121 (2011), 1565-1587.  doi: 10.1016/j.spa.2011.03.015.

[11]

R. DoucO. Cappé and E. Moulines, Comparison of resampling schemes for particle filtering, Proc. 4th Int. Symp. on Image and Signal Processing and Analysis, (2005), 64-69.  doi: 10.1109/ISPA.2005.195385.

[12]

A. Doucet, N. De Freitas and N. J. Gordon, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3437-9.

[13]

A. Ferreiro-CastillaA. E. KyprianouR. Scheichl and G. Suryanarayana, Multilevel Monte Carlo simulation for Lévy processes based on the Wiener-Hopf factorization, Stoch. Proc. Appl., 124 (2014), 985-1010.  doi: 10.1016/j.spa.2013.09.015.

[14]

M. P. S. Gander and D. A. Stephens, Simulation and inference for stochastic volatility models driven by Lévy processes, Biometrika, 94 (2007), 627-646.  doi: 10.1093/biomet/asm048.

[15]

M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.

[16]

M. B. Giles and Y. Xia, Multilevel path simulation for jump-diffusion SDEs, PMonte Carlo and Quasi-Monte Carlo methods 2010, Springer Proc. Math. Stat., Springer, Heidelberg, 23 (2012), 695-708.  doi: 10.1007/978-3-642-27440-4_41.

[17]

A. Golightly and D. J. Wilkinson., Bayesian inference for nonlinear multivariate diffusion models observed with error, Comp. Stat. Data Anal., 52 (2008), 1674-1693.  doi: 10.1016/j.csda.2007.05.019.

[18]

J. E. Griffin and M. F. J. Steel, Inference with non-Gaussian Ornstein-Uhlenbeck processes for stochastic volatility, J. Econom., 134 (2006), 605-644.  doi: 10.1016/j.jeconom.2005.07.007.

[19]

J. JacodT. G. KurtzS. Méléard and P. Protter, The approximate Euler method for Lévy driven stochastic differential equations, Ann. Inst. Henri Poincaré Probab. Stat., 41 (2005), 523-558.  doi: 10.1016/j.anihpb.2004.01.007.

[20]

A. JasraK. KamataniK. J. H. Law and Y. Zhou, Multilevel particle filters, SIAM J. Numer. Anal., 55 (2017), 3068-3096.  doi: 10.1137/17M1111553.

[21]

A. JasraK. KamataniK. J. H. Law and Y. Zhou, Bayesian static parameter estimation for partially observed diffusions via multilevel Monte Carlo, SIAM J. Sci. Comp., 40 (2018), A887-A902.  doi: 10.1137/17M1112595.

[22]

A. JasraK. Kamatani and H. Masuda, Bayesian inference for stable Lévy driven stochastic differential equations with high-frequency data, Scand. J. Stat., 46 (2019), 545-574.  doi: 10.1111/sjos.12362.

[23]

A. JasraK. J. H. Law and P. P. Osei, Multilevel particle filters for Lévy-driven stochastic differential equations, Stat. Comput., 29 (2019), 775-789.  doi: 10.1007/s11222-018-9837-z.

[24]

A. JasraD. A. StephensA. Doucet and T. Tsagaris, Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.  doi: 10.1111/j.1467-9469.2010.00723.x.

[25]

A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, Universitext, Springer, Berlin, Heidelberg, 2014. doi: 10.1007/978-3-642-37632-0.

[26]

A. E. Kyprianou, W. Schoutens and P. Wilmott, Exotic Option Pricing and Advanced Lévy Models, John Wiley & Sons, Ltd., Chichester, 2005.

[27]

D. McLeish, A general method for debiasing a Monte Carlo estimator, Monte Carlo Methods Appl., 17 (2011), 301-315.  doi: 10.1515/mcma.2011.013.

[28]

P. E. Protter, Stochastic Integration and Differential Equations, Second edition, Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.

[29]

P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25 (1997), 393-423.  doi: 10.1214/aop/1024404293.

[30]

C.-H. Rhee and P. W. Glynn, Unbiased estimation with square root convergence for SDE models, Oper. Res., 63 (2015), 1026-1043.  doi: 10.1287/opre.2015.1404.

[31]

G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm, Biometrika, 88 (2001), 603-621.  doi: 10.1093/biomet/88.3.603.

[32]

S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process, Stochastic Process. Appl., 103 (2003), 311-349.  doi: 10.1016/S0304-4149(02)00191-6.

[33]

H. M. Ruzayqat and A. Jasra, Unbiased estimation of the solution to Zakai's equation, Monte Carlo Methods Appl., 26 (2020), 113-129.  doi: 10.1515/mcma-2020-2061.

[34]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.

[35]

M. ViholaJ. Helske and J. Franks, Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo, Scand. J. Statist., 47 (2020), 1339-1376.  doi: 10.1111/sjos.12492.

show all references

References:
[1]

C. AndrieuA. Doucet and R. Holenstein, Particle Markov chain Monte Carlo methods, J. R. Stat. Soc. Ser. B Stat. Methodol., 72 (2010), 269-342.  doi: 10.1111/j.1467-9868.2009.00736.x.

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755323.

[3]

O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0197-7.

[4]

O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian OU-based models and some of their uses in financial economics, J. Roy. Statist. Soc. B Stat. Methodol., 63 (2001), 167-241.  doi: 10.1111/1467-9868.00282.

[5]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996.

[6]

R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech., 10 (1961), 493-516. 

[7]

O. Cappé, T. Ryden and E. Moulines, Inference in Hidden Markov Models, Springer, New York, 2005.

[8]

N. K. ChadaJ. FranksA. JasraK. J. H. Law and M. Vihola, Unbiased inference for discretely observed hidden markov model diffusions, SIAM/ASA J. Uncertain. Quantif., 9 (2021), 763-787.  doi: 10.1137/20M131549X.

[9]

P. Del Moral, Feyman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9393-1.

[10]

S. Dereich and F. Heidenreich, A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations, Stoc. Proc. Appl., 121 (2011), 1565-1587.  doi: 10.1016/j.spa.2011.03.015.

[11]

R. DoucO. Cappé and E. Moulines, Comparison of resampling schemes for particle filtering, Proc. 4th Int. Symp. on Image and Signal Processing and Analysis, (2005), 64-69.  doi: 10.1109/ISPA.2005.195385.

[12]

A. Doucet, N. De Freitas and N. J. Gordon, Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3437-9.

[13]

A. Ferreiro-CastillaA. E. KyprianouR. Scheichl and G. Suryanarayana, Multilevel Monte Carlo simulation for Lévy processes based on the Wiener-Hopf factorization, Stoch. Proc. Appl., 124 (2014), 985-1010.  doi: 10.1016/j.spa.2013.09.015.

[14]

M. P. S. Gander and D. A. Stephens, Simulation and inference for stochastic volatility models driven by Lévy processes, Biometrika, 94 (2007), 627-646.  doi: 10.1093/biomet/asm048.

[15]

M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.

[16]

M. B. Giles and Y. Xia, Multilevel path simulation for jump-diffusion SDEs, PMonte Carlo and Quasi-Monte Carlo methods 2010, Springer Proc. Math. Stat., Springer, Heidelberg, 23 (2012), 695-708.  doi: 10.1007/978-3-642-27440-4_41.

[17]

A. Golightly and D. J. Wilkinson., Bayesian inference for nonlinear multivariate diffusion models observed with error, Comp. Stat. Data Anal., 52 (2008), 1674-1693.  doi: 10.1016/j.csda.2007.05.019.

[18]

J. E. Griffin and M. F. J. Steel, Inference with non-Gaussian Ornstein-Uhlenbeck processes for stochastic volatility, J. Econom., 134 (2006), 605-644.  doi: 10.1016/j.jeconom.2005.07.007.

[19]

J. JacodT. G. KurtzS. Méléard and P. Protter, The approximate Euler method for Lévy driven stochastic differential equations, Ann. Inst. Henri Poincaré Probab. Stat., 41 (2005), 523-558.  doi: 10.1016/j.anihpb.2004.01.007.

[20]

A. JasraK. KamataniK. J. H. Law and Y. Zhou, Multilevel particle filters, SIAM J. Numer. Anal., 55 (2017), 3068-3096.  doi: 10.1137/17M1111553.

[21]

A. JasraK. KamataniK. J. H. Law and Y. Zhou, Bayesian static parameter estimation for partially observed diffusions via multilevel Monte Carlo, SIAM J. Sci. Comp., 40 (2018), A887-A902.  doi: 10.1137/17M1112595.

[22]

A. JasraK. Kamatani and H. Masuda, Bayesian inference for stable Lévy driven stochastic differential equations with high-frequency data, Scand. J. Stat., 46 (2019), 545-574.  doi: 10.1111/sjos.12362.

[23]

A. JasraK. J. H. Law and P. P. Osei, Multilevel particle filters for Lévy-driven stochastic differential equations, Stat. Comput., 29 (2019), 775-789.  doi: 10.1007/s11222-018-9837-z.

[24]

A. JasraD. A. StephensA. Doucet and T. Tsagaris, Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.  doi: 10.1111/j.1467-9469.2010.00723.x.

[25]

A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, Universitext, Springer, Berlin, Heidelberg, 2014. doi: 10.1007/978-3-642-37632-0.

[26]

A. E. Kyprianou, W. Schoutens and P. Wilmott, Exotic Option Pricing and Advanced Lévy Models, John Wiley & Sons, Ltd., Chichester, 2005.

[27]

D. McLeish, A general method for debiasing a Monte Carlo estimator, Monte Carlo Methods Appl., 17 (2011), 301-315.  doi: 10.1515/mcma.2011.013.

[28]

P. E. Protter, Stochastic Integration and Differential Equations, Second edition, Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.

[29]

P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25 (1997), 393-423.  doi: 10.1214/aop/1024404293.

[30]

C.-H. Rhee and P. W. Glynn, Unbiased estimation with square root convergence for SDE models, Oper. Res., 63 (2015), 1026-1043.  doi: 10.1287/opre.2015.1404.

[31]

G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm, Biometrika, 88 (2001), 603-621.  doi: 10.1093/biomet/88.3.603.

[32]

S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process, Stochastic Process. Appl., 103 (2003), 311-349.  doi: 10.1016/S0304-4149(02)00191-6.

[33]

H. M. Ruzayqat and A. Jasra, Unbiased estimation of the solution to Zakai's equation, Monte Carlo Methods Appl., 26 (2020), 113-129.  doi: 10.1515/mcma-2020-2061.

[34]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.

[35]

M. ViholaJ. Helske and J. Franks, Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo, Scand. J. Statist., 47 (2020), 1339-1376.  doi: 10.1111/sjos.12492.

Figure 1.  The blue, red and black curves correspond to the density of $ \mu^l $, the CDF and the inverse of the CDF, respectively
Figure 2.  The computational cost (in seconds) versus MSE for both the unbiased inference method in Algorithm 8 and PMMH algorithm in [1]
Table 1.  First and third columns are the expected values of $ \theta $ obtained by Algorithm 8 and PMMH, respectively. The second and fourth columns are the corresponding MSE and the fifth column is the ratio of the computational costs
$ \theta_{ub} $ $ \text{MSE}_{ub} $ $ \theta_{pmmh} $ $ \text{MSE}_{pmmh} $ $ \text{Cost}_{pmmh}/\text{Cost}_{ub} $
0.78446 1.5054 E-01 0.78748 1.3591 E-01 7.98
0.76129 3.9087 E-02 0.78309 5.4409 E-02 6.02
0.76089 5.4825 E-03 0.78069 3.9315 E-03 17.1
0.76183 5.3014 E-04 0.77012 5.4522 E-04 28.5
$ \theta_{ub} $ $ \text{MSE}_{ub} $ $ \theta_{pmmh} $ $ \text{MSE}_{pmmh} $ $ \text{Cost}_{pmmh}/\text{Cost}_{ub} $
0.78446 1.5054 E-01 0.78748 1.3591 E-01 7.98
0.76129 3.9087 E-02 0.78309 5.4409 E-02 6.02
0.76089 5.4825 E-03 0.78069 3.9315 E-03 17.1
0.76183 5.3014 E-04 0.77012 5.4522 E-04 28.5
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