Advanced

Estimation and Model Validation of Diffusion Processes

Lindström, Erik LU (2003)
Abstract
Estimation and Model Validation of Diffusion Processes



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... (More)
Estimation and Model Validation of Diffusion Processes



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 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. (Less)
Please use this url to cite or link to this publication:
author
supervisor
organization
publishing date
type
Thesis
publication status
published
subject
language
English
LU publication?
yes
id
4633e48b-8e31-4abe-84b1-e4204ee3387a (old id 753347)
date added to LUP
2008-01-04 14:14:31
date last changed
2016-09-19 08:45:16
@misc{4633e48b-8e31-4abe-84b1-e4204ee3387a,
  abstract     = {Estimation and Model Validation of Diffusion Processes<br/><br>
<br/><br>
Abstract<br/><br>
<br/><br>
The main motivation for this thesis is the need for estimation and model <br/><br>
validation of diffusion processes, i.e. stochastic processes satisfying <br/><br>
a stochastic differential equation driven by Brownian motion. This class <br/><br>
of stochastic processes is a natural extension of ordinary differential <br/><br>
equations to dynamic, stochastic systems.<br/><br>
<br/><br>
However Maximum Likelihood estimation of diffusion processes is in <br/><br>
general not feasible as the transition probability density in not <br/><br>
available in closed form. This problem is tackled in paper A, where an <br/><br>
approximative Maximum Likelihood estimator based on numerical solution <br/><br>
of the Fokker-Planck equation is presented.<br/><br>
<br/><br>
Closely connected to estimation is the problem of model validation. <br/><br>
Models are usually validated by testing dependence and distributional <br/><br>
properties of the residuals. A numerically stable algorithm for <br/><br>
calculating independent and identically distributed Gaussian residuals <br/><br>
for diffusion processes is introduced in paper B.<br/><br>
<br/><br>
Two other validation techniques, based on Gaussian approximations of the <br/><br>
system of stochastic differential equations, are described in paper C. <br/><br>
The approximation makes it possible to use filtering techniques to <br/><br>
calculate standardized residuals, which are tested for dependence using <br/><br>
lag dependent functions.<br/><br>
<br/><br>
Finally, a technique is introduced for identification of potential model <br/><br>
deficiencies using the estimated diffusion term. The deficiencies are <br/><br>
investigated by non-parametric regression using e.g. states, input <br/><br>
signals or time as explanatory variables.<br/><br>
<br/><br>
Keywords: Stochastic differential equations, Validation, Estimation, <br/><br>
Fokker-Planck equation, Lag Dependent Functions.},
  author       = {Lindström, Erik},
  language     = {eng},
  title        = {Estimation and Model Validation of Diffusion Processes},
  year         = {2003},
}