Estimation and Model Validation of Diffusion Processes
Erik Lindström
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2003
ISSN 1404028X

Abstract:

The main motivation for this thesis is the need for estimation and model
validation of diffusion processes, i.e. stochastic processes satisfying a
stochastic differential equation driven by Brownian motion. This class of
stochastic processes is a natural extension of ordinary differential equations
to dynamic, stochastic systems.


However Maximum Likelihood estimation of diffusion processes is in general
not feasible as the transition probability density in not available in closed
form. This problem is tackled in paper A, where an approximative Maximum
Likelihood estimator based on numerical solution of the FokkerPlanck equation
is presented.


Closely connected to estimation is the problem of model validation. Models
are usually validated by testing dependence and distributional properties
of the residuals. A numerically stable algorithm for calculating independent
and identically distributed Gaussian residuals for diffusion processes is
introduced in paper B.


Two other validation techniques, based on Gaussian approximations of the
system of stochastic differential equations, are described in paper C. The
approximation makes it possible to use filtering techniques to calculate
standardized residuals, which are tested for dependence using lag dependent
functions.


Finally, a technique is introduced for identification of potential model
deficiencies using the estimated diffusion term. The deficiencies are
investigated by nonparametric regression using e.g. states, input signals
or time as explanatory variables.



Keywords:

Stochastic differential equations, Validation, Estimation, FokkerPlanck
equation, Lag Dependent Functions.

