Introducing a new approach for modeling a given time series based on attributing any random variation to a jump...

When analyzing the data sampled at discrete times, one encounters successive discontinuities in the trajectory of the sampled time series, even if the underlying path is continuous. On the other hand, the distinction between discontinuities caused by finite sampling of continuous stochastic process and real discontinuities in the sample path is one of the main problems. Clues like these led us to the question: Is it possible to provide a model that treats any random variation in the data set as a jump event, regardless of whether the given time series is classified as diffusion or jump-diffusion processes? To address this question, we wrote a new stochastic dynamical equation, which includes a drift term and a combination of Poisson jump processes with different distributed sizes. In this article, we first introduce this equation in its simplest form including a drift term and a jump process, and show that such a jump-drift equation is able to describe the discrete time evolution of a diffusion process. Afterwards, we extend the modeling by considering more jump processes in the equation, which can be used to model complex systems with various distributed amplitudes. At each step, we also show that all the unknown functions and parameters required for modeling can be obtained non-parametrically from the measured time series.