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On periodic parameter identification in stochastic differential equations

  • Corresponding author: Shuai Lu

    Corresponding author: Shuai Lu 
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).
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  • 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.

    Mathematics Subject Classification: Primary: 65L09; Secondary: 60H10.

    Citation:

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  • Figure 1.  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

    Figure 2.  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

    Figure 3.  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

    Figure 4.  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

    Figure 5.  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)

    Figure 6.  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)

    Figure 7.  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

    Table 1.  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 $.
     | Show Table
    DownLoad: CSV

    Table 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
     | Show Table
    DownLoad: CSV

    Table 3.  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
     | Show Table
    DownLoad: CSV
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