Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

✍ Scribed by Sylvie Méléard; Sylvie Roelly

• Publisher: John Wiley and Sons
• Year: 1991
• Tongue: English
• Weight: 722 KB
• Volume: 154
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19911540112

✦ Synopsis

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function A are semimartingales whose martingale terms are identified with integrals off with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of Lz(Q)-valued vector measures.