| Title | Estimation and Model Validation of Diffusion Processes |
| Authors | Erik Lindström |
| Year | 2003 |
| Document type | Licentiate thesis |
| Language | eng |
| Abstract English | Estimation and Model Validation of Diffusion Processes<br> <br> Abstract<br> <br> The main motivation for this thesis is the need for estimation and model <br> validation of diffusion processes, i.e. stochastic processes satisfying <br> a stochastic differential equation driven by Brownian motion. This class <br> of stochastic processes is a natural extension of ordinary differential <br> equations to dynamic, stochastic systems.<br> <br> However Maximum Likelihood estimation of diffusion processes is in <br> general not feasible as the transition probability density in not <br> available in closed form. This problem is tackled in paper A, where an <br> approximative Maximum Likelihood estimator based on numerical solution <br> of the Fokker-Planck equation is presented.<br> <br> Closely connected to estimation is the problem of model validation. <br> Models are usually validated by testing dependence and distributional <br> properties of the residuals. A numerically stable algorithm for <br> calculating independent and identically distributed Gaussian residuals <br> for diffusion processes is introduced in paper B.<br> <br> Two other validation techniques, based on Gaussian approximations of the <br> system of stochastic differential equations, are described in paper C. <br> The approximation makes it possible to use filtering techniques to <br> calculate standardized residuals, which are tested for dependence using <br> lag dependent functions.<br> <br> Finally, a technique is introduced for identification of potential model <br> deficiencies using the estimated diffusion term. The deficiencies are <br> investigated by non-parametric regression using e.g. states, input <br> signals or time as explanatory variables.<br> <br> Keywords: Stochastic differential equations, Validation, Estimation, <br> Fokker-Planck equation, Lag Dependent Functions. |
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