NONDIFFERENTIABLE FUNCTIONS OF ONE-DIMENSIONAL SEMIMARTINGALES

• 成果类型: Article
• 署名作者: Lowther, George
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP476
• 发表日期: 2010
• 页码: 76-101
• 关键词: dirichlet processes; calculus

摘要

We consider decompositions of processes of the form Y = f (t, X(t)) where X is a semimartingale. The function f is not required to be differentiable, so Ito's lemma does not apply. In the case where f (t, x) is independent of t, it is shown that requiring f to be locally Lipschitz continuous in x is enough for an Ito-style decomposition to exist. In particular, Y will be a Dirichlet process. We also look at the case where f (t, x) can depend on t, possibly discontinuously. It is shown, under some additional mild constraints on f, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes. Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.