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Stochastic processes with jumps and non-vanishing higher-order kramers–moyal coefficients

Rahimi Tabar, M. R ; Sharif University of Technology | 2019

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  1. Type of Document: Article
  2. DOI: 10.1007/978-3-030-18472-8_11
  3. Publisher: Springer Verlag , 2019
  4. Abstract:
  5. In this chapter we study stochastic processes in the presence of jump discontinuity, and discuss the meaning of non-vanishing higher-order Kramers–Moyal coefficients. We describe in details the stochastic properties of Poisson jump processes. We derive the statistical moments of the Poisson process and the Kramers–Moyal coefficients for pure Poisson jump events. Growing evidence shows that continuous stochastic modeling (white noise-driven Langevin equation) of time series of complex systems should account for the presence of discontinuous jump components [1–6]. Such time series have some distinct important characteristics, such as heavy tails and occasionally sudden large jumps. Nonparametric (data-based) modeling of time series with jumps provides an attractive way of conducting research and gaining intuition of such processes. The focus in this chapter is introducing stochastic tools for investigation of time series with discontinuous jump components. We will start with the meaning of non-vanishing higher order KM coefficients and its impact on the continuity condition. Similarly to the role of the Wiener process in the Langevin modeling, the Poisson jump process plays an essential role in jump-diffusion modeling. Therefore we present stochastic properties of the Poisson jump process, such as its statistical moments, waiting time distribution, etc. © 2019, Springer Nature Switzerland AG
  6. Keywords:
  7. Continuity condition ; Jumps in time series ; Non-vanishing higher-order Kramers–Moyal coefficients ; Poisson jump process ; Statistical moments
  8. Source: Understanding Complex Systems ; 2019 , Pages 99-110 ; 18600832 (ISSN)
  9. URL: https://link.springer.com/chapter/10.1007%2F978-3-030-18472-8_11