Stochastic Equations of Super-L\'{e}vy Process with General Branching Mechanism

Speaker: 
Xu Yang
Date: 
Mon, Jun 18, 2012
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-MPrime Summer School in Probability
Abstract: 
The process of distribution functions of a one-dimensional super-L\'{e}vy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time-space white noises and Poisson random measures. This generalizes a recent result of Xiong (2012), where the result for a super-Brownian motion with binary branching mechanism was obtained. To establish the main result, we prove a generalized It\^o's formula for backward stochastic integrals and study the pathwise uniqueness for a general backward doubly stochastic equation with jumps. Furthermore, we also present some results on the SPDE driven by a one-sided stable noise without negative jumps. This is a joint work with Hui He and Zenghu Li.