# American Institute of Mathematical Sciences

June  2019, 13(3): 513-543. doi: 10.3934/ipi.2019025

## On periodic parameter identification in stochastic differential equations

 Shanghai Key Laboratory for Contemporary Applied Mathematics, Key Laboratory of Mathematics for Nonlinear Sciences and School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Shuai Lu

Received  April 2018 Revised  October 2018 Published  March 2019

Fund Project: S. Lu is supported by National Key Research and Development Program of China (No. 2017YFC1404103), NSFC (No.91730304, 11522108), Shanghai Municipal Education Commission (No.16SG01) and Special Funds for Major State Basic Research Projects of China (2015CB856003). J. Cheng is supported by NSFC (key projects no.11331004, no.11421110002) and the Programme of Introducing Talents of Discipline to Universities (number B08018)

Periodic parameters are common and important in stochastic differential equations (SDEs) arising in many contemporary scientific and engineering fields involving dynamical processes. These parameters include the damping coefficient, the volatility or diffusion coefficient and possibly an external force. Identification of these periodic parameters allows a better understanding of the dynamical processes and their hidden intermittent instability. Conventional approaches usually assume that one of the parameters is known and focus on the recovery of rest parameters. By introducing the decorrelation time and calculating the standard Gaussian statistics (mean, variance) explicitly for the scalar Langevin equations with periodic parameters, we propose a parameter identification approach to simultaneously recovering all these parameters by observing a single trajectory of SDEs. Such an approach is summarized in form of regularization schemes with noisy operators and noisy right-hand sides and is further extended to parameter identification of SDEs which are indirectly observed by other random processes. Numerical examples show that our approach performs well in stable and weakly unstable regimes but may fail in strongly unstable regime which is induced by the strong intermittent instability itself.

Citation: Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
##### References:
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show all references

##### References:
 [1] G. Bao, C. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016), 1263-1287.  doi: 10.1137/16M1067470.  Google Scholar [2] G. Bao, S. N. Chow and H. Zhou, An inverse random source problem for the Helmholtz equation, Mathematics of Computation, 83 (2014), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5.  Google Scholar [3] G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2013), 015006, 16 pp. doi: 10.1088/0266-5611/29/1/015006.  Google Scholar [4] G. Bao and X. Xu, Identification of the material properties in nonuniform nanostructures, Inverse Problems, 31 (2015), 125003, 11 pp. doi: 10.1088/0266-5611/31/12/125003.  Google Scholar [5] M. Branicki and A. J. Majda, Quantifying Bayesian filter performance for turbulent dynamical systems through information theory, Commun. Math. Sci., 12 (2014), 901-978.  doi: 10.4310/CMS.2014.v12.n5.a6.  Google Scholar [6] D. G. Cacuci, I. M. Navon and M. Ionescu-Bujor, Computational Methods for Data Evaluation and Assimilation, CRC Press, Boca Raton, FL, 2014.  Google Scholar [7] N. Chen, D. Giannakis, R. Herbei and A. J. Majda, An MCMC algorithm for parameter estimation in signals with hidden intermittent instability, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 647-669.  doi: 10.1137/130944977.  Google Scholar [8] M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inverse Problems, 33 (2017), 095006, 12 pp. doi: 10.1088/1361-6420/aa7a1c.  Google Scholar [9] F. Dunker and T. Hohage, On parameter identification in stochastic differential equations by penalized maximum likelihood, Inverse Problems, 30 (2014), 095001, 20 pp. doi: 10.1088/0266-5611/30/9/095001.  Google Scholar [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. viii+321 pp.  Google Scholar [11] H. Gao, K. Wang, F. Wei and X. Ding, Massera-type theorem and asymptotically periodic logistic equations, Nonlinear Analysis: Real World Applications, 7 (2006), 1268-1283.  doi: 10.1016/j.nonrwa.2005.11.008.  Google Scholar [12] B. Gershgorin, J. Harlim and A. J. Majda, Test models for improving filtering with model errors through stochastic parameter estimation, Journal of Computational Physics, 229 (2010), 1-31.  doi: 10.1016/j.jcp.2009.08.019.  Google Scholar [13] A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation, Biometrics, 61 (2005), 781-788.  doi: 10.1111/j.1541-0420.2005.00345.x.  Google Scholar [14] B. Kaltenbacher and B. Pedretscher, Parameter estimation in SDEs via the Fokker-Planck equation: Likelihood function and adjoint based gradient computation, Journal of Mathematical Analysis and Applications, 465 (2018), 872-884.  doi: 10.1016/j.jmaa.2018.05.048.  Google Scholar [15] W. J. Lee and A. Stuart, Derivation and analysis of simplified filters, Communications in Mathematical Sciences, 15 (2017), 413-450.  doi: 10.4310/CMS.2017.v15.n2.a6.  Google Scholar [16] P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004, 22 pp. doi: 10.1088/0266-5611/27/3/035004.  Google Scholar [17] S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems: Selected Topics, Inverse and Ill-posed Problems Series, 58 De Gruyter, Berlin, 2013. doi: 10.1515/9783110286496.  Google Scholar [18] S. Lu, S. V. Pereverzev, Y. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, Journal of Integral Equations and Application, 22 (2010), 483-517.  doi: 10.1216/JIE-2010-22-3-483.  Google Scholar [19] A. J. Majda and J. Harlim, Filtering Complex Turbulent Systems, Cambridge University Press, Cambridge, 2012. x+357 pp. doi: 10.1017/CBO9781139061308.  Google Scholar [20] A. J. Majda and X. Wang, Non-linear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511616778.  Google Scholar [21] B. Øksendal, Stochastic Differential Equations, 6$^{th}$ ed., Springer, Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [22] O. Papaspiliopoulos, Y Pokern, G. O. Roberts and A. M. Stuart, Nonparametric estimation of diffusions: A differential equations approach, Biometrika, 99 (2012), 511-531.  doi: 10.1093/biomet/ass034.  Google Scholar [23] G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion model using Metropolis–Hastings algorithms, Biometrika, 88 (2001), 603-621.  doi: 10.1093/biomet/88.3.603.  Google Scholar [24] H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: A survey, International Statistical Review, 72 (2004), 337-354.  doi: 10.1111/j.1751-5823.2004.tb00241.x.  Google Scholar [25] O. Stramer and M. Bognar, Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach, Bayesian Analysis, 6 (2011), 231-258.  doi: 10.1214/11-BA608.  Google Scholar [26] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar [27] U. Tautenhahn, Regularization of linear ill-posed problems with noisy right hand side and noisy operator, J. Inv. Ill-Posed Problems, 16 (2008), 507-523.  doi: 10.1515/JIIP.2008.027.  Google Scholar [28] C. R. Vogel, Computational Methods for Inverse Problems, With a foreword by H. T. Banks. Frontiers in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.  Google Scholar
Pathwise solutions of the stochastic differential equation (4) with different parameters in Table 1. Upper (stable regime); middle (weakly unstable regime) and bottom (strongly unstable regime). Each panel presents a segment of $v(t)$ for $t\in [100,102]$ whereas the long path of the solution $v(t)$ for $t\in [0,20000]$ is presented in the small picture in each panel
Empirical values of $5$ trajectories of the random process $v(t)$ in (4) with different parameters in Table 1. Upper row (stable regime); middle row (weakly unstable regime) and bottom row (strongly unstable regime). The red solid line in each panel is the exact value. The blue dashed lines are empirical values of $5$ different trajectories
Pathwise solutions of the stochastic differential equations (23)-(24) with different parameters in Table 1 and (30). Upper row: $v(t)$ and $u(t)$ (stable regime); middle row: $v(t)$ and $u(t)$ (weakly unstable regime) and bottom row: $v(t)$ and $u(t)$ (strongly unstable regime). Each panel presents a segment of $v(t)$ or $u(t)$ for $t\in [100,102]$ whereas the long path of the solutions for $t\in [0,40000]$ is presented in the small picture in each panel
Empirical values of $5$ trajectories of the random process $u(t)$ in (23)-(24) with different parameters in Table 1 and (30). Upper row (stable regime); middle row (weakly unstable regime) and bottom row (strongly unstable regime). The red solid line in each panel is the exact value. The blue dashed lines are empirical values of $5$ different trajectories
Parameter identification approach for direct observation $v(t)$ in (4). Upper row: exact and reconstructed $\gamma_v$, $f_{v}$ and $\sigma_v^2$ (stable regime); middle row: exact and reconstructed $\gamma_v$, $f_{v}$ and $\sigma_v^2$ (weakly unstable regime) and bottom row: exact and reconstructed $\gamma_v$, $f_{v}$ and $\sigma_v^2$ (strongly unstable regime). Small figures in each panel are the exact (red solid line) and empirical (blue dashed line) decorrelation time, mean and variance of $v(t)$ which are used in the parameter identification approach (20)
Parameter identification approach for indirect observation $u(t)$ in (23)-(24). Upper row: exact and reconstructed $\gamma_v$, $f_{v}$ and $\sigma_v^2$ (stable regime); middle row: exact and reconstructed $\gamma_v$, $f_{v}$ and $\sigma_v^2$ (weakly unstable regime) and bottom row: exact and reconstructed $\gamma_v$, $f_{v}$ and $\sigma_v^2$ (strongly unstable regime). Small figures in each panel are the exact (red solid line) and empirical (blue dashed line) decorrelation time, mean and variance of $u(t)$ which are used in the parameter identification approach (28)
Empirical values of $5$ trajectories of the random process $v(t)$ (upper row) and $u(t)$ (bottom row) with reconstructed parameters of the strongly unstable regime. The red solid line in each panel is the exact Gaussian statistics. The blue dashed lines are empirical Gaussian statistics of $5$ different trajectories by the reconstructed parameters
Parameters of the stochastic differential equations (4) and (23)
 Parameters Value Damping parameter (stable regime): $\gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+2.05$ (weakly unstable regime): $\gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+1.9$ (strongly unstable regime): $\gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+0.05$ Force parameter $f_v(\zeta)=0.1\sin(4\pi\zeta)+0.2$ Volatility parameter $\sigma_v^2(\zeta)=\left(0.1\sin(2\pi\zeta)+0.3\right)^2$.
 Parameters Value Damping parameter (stable regime): $\gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+2.05$ (weakly unstable regime): $\gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+1.9$ (strongly unstable regime): $\gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+0.05$ Force parameter $f_v(\zeta)=0.1\sin(4\pi\zeta)+0.2$ Volatility parameter $\sigma_v^2(\zeta)=\left(0.1\sin(2\pi\zeta)+0.3\right)^2$.
Direct observation. Columns 2-4: $L^2-$relative errors of exact and empirical Gaussian statistics of $v(t)$. Final three columns: $L^2-$relative errors of the exact and reconstructed parameters. The observation time is $[15000,20000]$
 Decorrelation time Mean Variance $\gamma_v$ $f_v$ $\sigma_v^2$ stable 0.0287 0.0170 0.0173 0.0347 0.0706 0.0271 weakly unstable 0.0347 0.0295 0.0173 0.0601 0.0949 0.0662 strongly unstable 0.0049 0.0274 0.0310 0.0178 0.5762 0.3591
 Decorrelation time Mean Variance $\gamma_v$ $f_v$ $\sigma_v^2$ stable 0.0287 0.0170 0.0173 0.0347 0.0706 0.0271 weakly unstable 0.0347 0.0295 0.0173 0.0601 0.0949 0.0662 strongly unstable 0.0049 0.0274 0.0310 0.0178 0.5762 0.3591
Indirect observation. Columns 2-4: $L^2-$relative errors of exact and empirical Gaussian statistics of $u(t)$. Final three columns: $L^2-$relative errors of the exact and reconstructed parameters. The observation time is $[30000,40000]$
 Decorrelation time Mean Variance $\gamma_v$ $f_v$ $\sigma_v^2$ stable 0.0108 0.0043 0.0098 0.0496 0.0916 0.0613 weakly unstable 0.0066 0.0099 0.0151 0.0208 0.1724 0.0773 strongly unstable 0.0090 0.0142 0.0575 0.0187 0.3866 0.1462
 Decorrelation time Mean Variance $\gamma_v$ $f_v$ $\sigma_v^2$ stable 0.0108 0.0043 0.0098 0.0496 0.0916 0.0613 weakly unstable 0.0066 0.0099 0.0151 0.0208 0.1724 0.0773 strongly unstable 0.0090 0.0142 0.0575 0.0187 0.3866 0.1462
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