STRONG SUPERMARTINGALES AND LIMITS OF NONNEGATIVE MARTINGALES

• 成果类型: Article
• 署名作者: Czichowsky, Christoph; Schachermayer, Walter
• 署名单位: University of London; London School Economics & Political Science; University of Vienna
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP970
• 发表日期: 2016
• 页码: 171-205
• 关键词: theorem; INVESTMENT; MARKETS

摘要

Given a sequence (M-n)(n=1)(infinity) of nonnegative martingales starting at M-0(n) = 1, we find a sequence of convex combinations (M-n)(n=1)(infinity) and a limiting process X such that (M-n)(n=1)(infinity) converges in probability to X-tau, for all finite stopping times tau. The limiting process X then is an optional strong supermartingale. A counterexample reveals that the convergence in probability cannot be replaced by almost sure convergence in this statement. We also give similar convergence results for sequences of optional strong super martingales (X-n)(n=1)(infinity), their left limits (X--(n))(n=1)(infinity) and their stochastic integrals (integral phi dX(n))(n=1)(infinity) and explain the relation to the notion of the Fatou limit.