On L2 projections on a space of stochastic integrals

• 成果类型: Article
• 署名作者: Rheinlander, T; Schweizer, M
• 署名单位: Technical University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1023481112
• 发表日期: 1997
• 页码: 1810-1831
• 关键词: martingale measure

摘要

Let X be an R-d-valued continuous semimartingale, T a fixed time horizon and Theta the space of all R-d-valued predictable X-integrable processes such that the stochastic integral G(theta) = integral theta dX is a square-integrable semimartingale. A recent paper gives necessary and sufficient conditions on X for G(T)(Theta) to be closed in L-2(P). In this paper, we describe the structure of the L-2-projection mapping an F-T-measurable random variable H is an element of L-2(P) on GT(Theta) and provide the resulting integrand theta(H) is an element of Theta in feedback form. This is related to variance-optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on X. Our proofs use the variance-optimal martingale measure (Pq) over tilde for X and weighted norm inequalities relating (P) over tilde to the original measure P.