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Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts

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  • Martin Friesen
  • Stefan Gerhold
  • Kristof Wiedermann

Abstract

We study small-time central limit theorems for stochastic Volterra integral equations with H\"older continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts.

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  • Martin Friesen & Stefan Gerhold & Kristof Wiedermann, 2024. "Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts," Papers 2412.15971, arXiv.org.
    • Handle: RePEc:arx:papers:2412.15971
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    File URL: http://arxiv.org/pdf/2412.15971
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