PATHWISE INEQUALITIES FOR LOCAL TIME: APPLICATIONS TO SKOROKHOD EMBEDDINGS AND OPTIMAL STOPPING

• 成果类型: Article
• 署名作者: Cox, A. M. G.; Hobson, David; Obloj, Jan
• 署名单位: University of Bath; University of Warwick; Imperial College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP507
• 发表日期: 2008
• 页码: 1870-1896
• 关键词: martingales

摘要

We develop a class of pathwise inequalities of the form H(B-t) >= M-t + F(L-t), where B-t is Brownian motion, L-t its local time at zero and M-t a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form sup(tau) E[F(L-tau) - integral(t)(0) beta(B-s)ds]. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.