A BDG inequality for stochastic Volterra integrals
Alexandre Pannier
TL;DR
The paper develops Burkholder-Davis-Gundy-type inequalities for stochastic Volterra integrals with completely monotone kernels, establishing finite-horizon bounds with constants scaling like $\|K\|_{L^\gamma([0,T])}$ for any $\gamma>2$ and $p>\tfrac{2\gamma}{\gamma-2}$. It leverages a convolution representation to prove these results and then applies them to stochastic Volterra equations, enabling precise pathwise comparisons between SVEs with different kernels and providing shift and multifactor kernel-approximation schemes with explicit convergence rates. An infinite-horizon (uniform-in-time) bound is provided under sufficient kernel decay, with applications to mean-reverting SVEs. The note also situates the new results within the landscape of classical BDG inequalities (Decreusefond; Kolmogorov continuity) and discusses their implications for rough volatility models and kernel-based approximations. Overall, the work supplies practical, verifiable tools for analyzing SVEs with memory and for designing kernel-approximation schemes in computational settings.
Abstract
We establish Burkholder-Davis-Gundy-type inequalities for stochastic Volterra integrals with a completely monotone convolution kernel, which may exhibit singular behaviour at the origin. When the supremum is taken over a finite interval, the upper bound depends linearly on the $L^γ$-norm of the kernel, for any $γ>2$. We demonstrate the utility of this inequality in quantifying the pathwise distance between two stochastic Volterra equations with distinct kernels, with a particular emphasis on the multifactor Markovian approximation. For kernels that decay sufficiently fast, we derive an alternative inequality valid over an infinite time interval, providing uniform-in-time bounds for mean-reverting stochastic Volterra equations. Finally, we compare our findings with existing results in the literature.