Weak $L^p$ inequalities for stochastic integrals with respect to random measures

Ivan Yaroslavtsev (U Hamburg)

Feb 13. 2024, 16:00 — 16:40

It is known due to Gin\'e and Marcus (1983) and Rosi\'nski and Woyczy\'nski (1986) that for any $\alpha \in (0,2)$, for any symmetric $\alpha$-stable L\'evy process $L$, and for any predictable real-valued process $\phi$ one has that
\sup_{\lambda>0}\lambda^{\alpha}\mathbb P \Bigl(\sup_{t\geq 0}\Bigl|\int_0^{t}\phi_s d L_s\Bigr|>\lambda\Bigr) \eqsim_{L}\mathbb E \int_0^{\infty}|\phi_s|^{\alpha} d s,
i.e.\ the weak $L^{\alpha}$-norm of the integral can be controlled by the $L^{\alpha}$-norm of the integrand.

The goal of the talk is to extend this result to both random measure setting and Banach space-valued integrals.

The talk is based on an ongoing work with Gergely Bod\'o (Universiteit van Amsterdam).

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Stochastic Partial Differential Equations (Workshop)
Sandra Cerrai (U of Maryland)
Martin Hairer (Imperial College London)
Carlo Marinelli (University College London)
Eulalia Nualart (U Barcelona)
Luca Scarpa (Politecnico Milano)
Ulisse Stefanelli (U of Vienna)