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Functional CLTs for subordinated L\'evy models in physics, finance, and econometrics

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  • Andreas S{o}jmark
  • Fabrice Wunderlich

Abstract

We present a simple unifying treatment of a broad class of applications from statistical mechanics, econometrics, mathematical finance, and insurance mathematics, where (possibly subordinated) L\'evy noise arises as a scaling limit of some form of continuous-time random walk (CTRW). For each application, it is natural to rely on weak convergence results for stochastic integrals on Skorokhod space in Skorokhod's J1 or M1 topologies. As compared to earlier and entirely separate works, we are able to give a more streamlined account while also allowing for greater generality and providing important new insights. For each application, we first elucidate how the fundamental conclusions for J1 convergent CTRWs emerge as special cases of the same general principles, and we then illustrate how the specific settings give rise to different results for strictly M1 convergent CTRWs.

Suggested Citation

  • Andreas S{o}jmark & Fabrice Wunderlich, 2023. "Functional CLTs for subordinated L\'evy models in physics, finance, and econometrics," Papers 2312.15119, arXiv.org, revised Jan 2024.
    • Handle: RePEc:arx:papers:2312.15119
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    5. Scalas, Enrico & Viles, Noèlia, 2014. "A functional limit theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 385-410.
    6. Caner, Mehmet, 1997. "Weak Convergence to a Matrix Stochastic Integral with Stable Processes," Econometric Theory, Cambridge University Press, vol. 13(4), pages 506-528, February.
    7. S. James Press, 1967. "A Compound Events Model for Security Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 317-317.
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