On uniqueness of solutions to stochastic equations: A counter-example

• 成果类型: Article
• 署名作者: Engelbert, HJ
• 署名单位: Friedrich Schiller University of Jena
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1039-1043
• 关键词: differential-equations

摘要

We consider the one-dimensional stochastic equation X-t = X-0 + integral(0)(t) b(X-s) dM(s) where M is a continuous local martingale and b a measurable real function. Suppose that b(-2) is locally integrable. D. N. Hoover asserted that, on a saturated probability space, there exists a solution X of the above equation with X-0 = 0 having no occupation time in the zeros of b and, moreover, the pair (X, M) is unique in law for all such X. We will give an example which shows that the uniqueness assertion fails, in general.