Sammanfattning
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 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.
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.
Originalspråk  engelska 

Kvalifikation  Licentiat 
Tilldelande institution 

Handledare 

Status  Published  2003 
Ämnesklassifikation (UKÄ)
 Sannolikhetsteori och statistik