Premium
Discontinuous Measure‐Valued Branching Processes and Generalized Stochastic Equations
Author(s) -
Méléard Sylvie,
Roelly Sylvie
Publication year - 1991
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19911540112
Subject(s) - mathematics , martingale (probability theory) , local martingale , martingale difference sequence , lévy process , measure (data warehouse) , stochastic differential equation , mathematical analysis , integrable system , pure mathematics , stochastic partial differential equation , differential equation , database , computer science
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 f , are semimartingales whose martingale terms are identified with integrals of f 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 L 2 (Ω)‐valued vector measures.