Feller processes: the next generation in modeling. Brownian motion, Lévy processes and beyond

Area of Science:

• Stochastic Processes
• Mathematical Finance
• Computational Statistics

Background:

• Brownian Motion and Lévy processes are widely used continuous-time Markov processes.
• Empirical data often exhibit spatial inhomogeneity, necessitating more general models than homogeneous Lévy processes.
• Feller processes offer a generalization that includes Lévy processes and Brownian Motion while accommodating spatial variations.

Purpose of the Study:

• To present a simple construction method for Feller processes.
• To establish a practical framework for generating sample paths of Feller processes.
• To enable the application of Monte Carlo methods to Feller processes for practitioners.

Main Methods:

• Construction of Feller processes via state-space-dependent mixing of Lévy processes.
• Development of a simulation technique for generating sample paths.
• Explanation of methods tailored for practical application and overcoming technical existence proofs.

Main Results:

• A simplified construction for Feller processes is provided.
• A framework for generating sample paths of Feller processes is established.
• The proposed simulation technique facilitates Monte Carlo analysis of Feller processes.

Conclusions:

• The presented method simplifies the theoretical and practical challenges associated with Feller processes.
• This work bridges the gap between theoretical properties and practical simulation of Feller processes.
• The framework empowers researchers and practitioners to utilize Feller processes in applications requiring spatial inhomogeneity.