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Limit theorems for local times and applications to SDEs with jumps

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  • Mijatović, Aleksandar
  • Uribe Bravo, Gerónimo

Abstract

Consider a stochastic process X, regenerative at a state x which is instantaneous and regular. Let L be a regenerative local time for X at x. Suppose furthermore that X can be approximated by discrete time regenerative processes Xn for which x is accessible. We give conditions on X and Xn so that the naturally defined local time of Xn (which counts the quantity of visits to x) converges weakly to L. This limit theorem generalizes previous invariance principles that have appeared in the literature. Furthermore, it allows one to prove novel invariance principles for local times of regenerative processes converging to diffusions or to solutions of SDEs with jumps.

Suggested Citation

  • Mijatović, Aleksandar & Uribe Bravo, Gerónimo, 2022. "Limit theorems for local times and applications to SDEs with jumps," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 39-56.
    • Handle: RePEc:eee:spapps:v:153:y:2022:i:c:p:39-56
    DOI: 10.1016/j.spa.2022.06.022
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    References listed on IDEAS

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    1. Kallenberg, Olav, 1992. "Some time change representations of stable integrals, via predictable transformations of local martingales," Stochastic Processes and their Applications, Elsevier, vol. 40(2), pages 199-223, March.
    2. Amaury Lambert & Florian Simatos, 2015. "Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case," Journal of Theoretical Probability, Springer, vol. 28(1), pages 41-91, March.
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