  In this study, we propose a new method that is useful for estimating unknown
parameter values of stochastic differential equation (SDE) models, based on
probability density function (PDF) data measured from random dynamical systems.
As our method does not require explicit description of PDF, it can be applied
to the SDE models even when their PDFs are hardly derived in explicit forms due
to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our
method is expected to provide a versatile tool to dynamically parameterize
measured PDF data. In our proposed method, it is assumed that a measured PDF is
obtained from a random dynamical system whose structure is described by a known
SDE model with unknown parameter values. With the help of It\^o calculus, the
Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF
and a candidate of parameter values are substituted into the FPE to calculate a
FPE residual. Our method is applied to two random vibration systems. Their FPE
residuals tend to zero as the parameter values tend to exact values, showing
that our proposed FPE residual can be utilized for unknown parameter estimation
of SDE models.
