MONOTONE STABILITY OF QUADRATIC SEMIMARTINGALES WITH APPLICATIONS TO UNBOUNDED GENERAL QUADRATIC BSDES

• 成果类型: Article
• 署名作者: Barrieu, Pauline; El Karoui, Nicole
• 署名单位: University of London; London School Economics & Political Science; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP743
• 发表日期: 2013
• 页码: 1831-1863
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; risk-sensitive control; Utility maximization; terminal conditions; incomplete markets; convex generators; GROWTH; valuation; entropy

摘要

In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential sub-martingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from H-1 to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied in particular to BSDE-like semimartingales. This strong convergence result is then used to prove the existence of solutions of general quadratic BSDEs under minimal exponential integrability assumptions, relying on a regularization in both linear-quadratic growth of the quadratic coefficient itself. On the contrary to most of the existing literature, it does not involve the seminal result of Kobylanski [Ann. Probab. 28 (2010) 558-602] on bounded solutions.