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
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Stochastic Partial Differential Equations (Workshop)
Organizer(s):
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)