Estimation and Model Validation of Diffusion Processes

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 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.