On a robust version of the integral representation formula of nonlinear filtering

• 成果类型: Article
• 署名作者: Clark, JMC; Crisan, D
• 署名单位: Imperial College London; Imperial College London
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0412-5
• 发表日期: 2005
• 页码: 43-56
• 关键词: equations

摘要

The paper is concerned with completing unfinished business on a robust representation formula for the conditional expectation operator of nonlinear filtering. Such a formula, robust in the sense that its dependence on the process of observations is continuous, was stated in [2] without proof. The main purpose of this paper is to repair this deficiency. The formula is almost obvious as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance. The continuity of the representation (but not its validity) was proved by Kushner [9] for a class of diffusions.