| dc.description.abstract | In this thesis we analyze the properties of a broad class of non-Markovian stochastic processes driven by Generalized Shot Noise (GSN) and aim to find their transition probabilities via distinct yet fundamental approaches. Stochastic processes are widely used mathematical tools to model uncertain behavior in a physical system. Recent studies show that the current stochastic models, which assume Markov or memoryless property, are inaccurate to model complex physical systems, from molecular dynamics in porous media to mild correlations of stock price returns in financial markets. We show that one can model these problems with a non-Markovian stochastic process X as the solution of the Generalized Langevin Equation (GLE) X˙ t = −V ′ (Xt) + ξt , where V is the potential of the environment and ξt is a GSN trajectory. We show that the non-Markovian GLE can be obtained by relaxing the Markov property with an impulse function h that creates diverse mathematical properties. Next, we show three distinct methods, each with their own rights and caveats, of finding the transition probability of Xt : first by directly from its characteristic functional, second by evaluating its time evolution equation, and third by formulating its path integral. Subsequently, we reserved the last part of our thesis for the application of the path integral results to two separate physical systems: tracking the position of a particle in a porous biological medium, and forecasting the price trajectory of a financial instrument that shows correlations in returns. | en_US |
