On periodic parameter identification in stochastic differential equations

Pingping Niu,Jin Cheng,Shuai Lu

doi:10.3934/ipi.2019025

Abstract

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.

Highlights

Stochastic differential equations (SDEs) arise in many contemporary scientific and engineering fields involving dynamical processes [21]
We show that the unknown parameters can be identified by minimizing the Gaussian statistics of mean, variance and decorrelation time
We show the performance of the parameter identification approach (20) for direct observation of v(t)

Summary

Introduction

Stochastic differential equations (SDEs) arise in many contemporary scientific and engineering fields involving dynamical processes [21]. We show that the unknown parameters can be identified by minimizing the Gaussian statistics of mean, variance and decorrelation time 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 another random process, i.e. Similar to Proposition 1, mean and variance of u(t) are asymptotically periodic whose forms are Fredholm integral equations with kernel functions depending on the damping parameters γu and γv. Since γu is assumed to be known, we need to reveal the unknown parameter γv(t) from the asymptotic decorrelation time of u(t) the same as in previous section To this end, we incorporate the asymptotical periodicity of mean and variance in Proposition 2 and define auxiliary functions (Fγu,γv fv )(ζ ). Parameter identification approach for the direct observation (stable regime) γv (ζ) 6

Exact fv Reconstructed fv

Conclusion and future plans

Estimation of the third term is more sophisticated and presented below

We thus prove