THE TRANSFORMATION THEOREM FOR 2-PARAMETER PURE JUMP MARTINGALES

• 成果类型: Article
• 署名作者: IMKELLER, P
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01198787
• 发表日期: 1991
• 页码: 261-283
• 关键词: STOCHASTIC INTEGRALS; REGULARITY; plane

摘要

Let M be a martingale of pure jump type, i.e. the compensation of the process describing the total of the point jumps of M in the plane. M can be uniformly approximated by martingales of bounded variation jumping only on finitely many axial parallel lines. Using this fact we prove a change of variables formula in which for C4-functions f the process f(M) is described by integrals of f(k)(M), k = 1,2, with respect to stochastic integrators of the types expected: a martingale, two processes behaving as martingales in one direction and as processes of bounded variation in the other, and one process of bounded variation. Hereby we are led to investigate two types of random measures not considered so far in this context. By combination with the integrators already known, they might complete the set needed for a general transformation formula.