Estimation and Model Validation of Diffusion Processes

Forskningsoutput: AvhandlingLicentiatavhandling


Estimation and Model Validation of Diffusion Processes


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 Fokker-Planck 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 non-parametric regression using e.g. states, input
signals or time as explanatory variables.

Keywords: Stochastic differential equations, Validation, Estimation,
Fokker-Planck equation, Lag Dependent Functions.
Tilldelande institution
  • Matematisk statistik
  • Holst, Jan, handledare
StatusPublished - 2003

Ämnesklassifikation (UKÄ)

  • Sannolikhetsteori och statistik


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