Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

3 Theory3.12 Artificial Test Examples for Hidden Markov Model -Vector Auto Regression (HMM-VAR)In order to test the HMM-VAR tools, developed and presented in section 3.11, it is advisable togenerate artificial trajectory data with a predefined set of parameters and afterwards estimate theseparameters by the HMM-VAR tools.We generated computationally different trajectories, two- and three dimensional, based on stochasticdifferential equations driven by a Gaussian white noise process. These trajectories are diffusionprocesses with or without drift. This means that we created trajectories, representing free diffusionon the one hand and anomalous diffusion (diffusion with an external potential force) on the otherhand.The trajectories are accomplished just in the framework of an HMM-SDE model, presented insection 3.10 on page 37, and mathematically represented by the equation 3.68 on page 38.In the next subsection we will show some results of the three-dimensional trajectory-generationprocess graphically, without estimating the parameters for these trajectories via HMM-VAR. Afterthis we turn to the two-dimensional trajectories and the problem of the parameter estimation viathe HMM-VAR tools (subsection 3.12.2).3.12.1 Three-Dimensional Diffusion TrajectoriesWe generated three-dimensional diffusion data, based on the HMM-SDE framework (see section3.10 on page 37), free diffusion as well as diffusion with drift, produced by a harmonicpotential. The results are shown in the Figures 3.9 and 3.10 for the free diffusion and Figure 3.11for the diffusion with potential.46