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

2024/12/20 by Friesen, Martin, Gerhold, Stefan, Wiedermann, Kristof
#60F05 #60F17 #60G15 #60G22 #60H15 #60H20 #91G20 #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.15971

Abstract

We study small-time central limit theorems for stochastic Volterra integral equations with Hölder 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.

Related

• Limit theorems for stochastic Volterra processes
• Markovian approximations of stochastic Volterra equations with the fractional kernel
• Small time central limit theorems for semimartingales with applications
• The central limit theorem for stochastic Volterra equations with singular kernels
• On a Stationarity Theory for Stochastic Volterra Integral Equations