Extracting stochastic dynamical systems with $\alpha$-stable L\'evy noise from data

From this point of view, dynamical modeling requires a deep understanding of the process to be analyzed. The essence of model abstraction is an approximation to the observed reality, which is usually represented by a system composed of ordinary or partial differential equations, deterministic or stochastic differential equations, and control equations. Although mathematical models are accurate for many processes, it is particularly difficult to develop such models for some of the most challenging systems, including climate dynamics, brain dynamics, biological systems and financial markets. Fortunately, more and more data are observed or measured in recent years with the development of scientific tools and simulation capabilities. Therefore, a large number of data-driven methods has been proposed to discover governing laws of systems from data. For instance, several researchers designed the Sparse Identification of Nonlinear Dynamics approach to extract deterministic ordinary [5] or partial [15, 29, 31] differential equations from available path data.