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Short-time behavior of solutions to Lévy-driven stochastic differential equations

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  12 January 2023

Jana Reker
Jana Reker*
Affiliation:
Ulm University
*
*Postal address:Ulm University, Institute of Mathematical Finance, Helmholtzstraße 18, 89081 Ulm, Germany. Email: J.Reker@RekerNet.de
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Abstract

We consider solutions of Lévy-driven stochastic differential equations of the form $\textrm{d} X_t=\sigma(X_{t-})\textrm{d} L_t$, $X_0=x$, where the function $\sigma$ is twice continuously differentiable and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix valued. While the almost sure short-time behavior of Lévy processes is well known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the solution X. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the form $t^{-p}\int_{0+}^t\sigma(X_{t-})\,\textrm{d} L_t$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely reflects the behavior of $t^{-p}L_t$. Generalizing $t^{{\kern1pt}p}$ to a suitable function $f\colon[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit law of the iterated logarithm type results for the solution from the behavior of the driving Lévy process.

Keywords

Short-time behaviorLévy-driven SDELIL-type results

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G51: Processes with independent increments; L'evy processes 60H20: Stochastic integral equations
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 765 - 780
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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