Operator Fitting for Parameter Estimation of Stochastic Differential
Equations

Estimation of parameters is a crucial part of model development. When models
are deterministic, one can minimise the fitting error; for stochastic systems
one must be more careful. Broadly parameterisation methods for stochastic
dynamical systems fit into maximum likelihood estimation- and method of
moment-inspired techniques. We propose a method where one matches a finite
dimensional approximation of the Koopman operator with the implied Koopman
operator as generated by an extended dynamic mode decomposition approximation.
One advantage of this approach is that the objective evaluation cost can be
independent the number of samples for some dynamical systems. We test our
approach on two simple systems in the form of stochastic differential
equations, compare to benchmark techniques, and consider limited
eigen-expansions of the operators being approximated. Other small variations on
the technique are also considered, and we discuss the advantages to our
formulation.