Harmonic analysis of stochastic equations and backward stochastic differential equations

• 成果类型: Article
• 署名作者: Delbaen, Freddy; Tang, Shanjian
• 署名单位: Fudan University; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0191-5
• 发表日期: 2010
• 页码: 291-336
• 关键词: follmer-schweizer decomposition; random-coefficients; quadratic growth; integral-equations; riccati-equations; adapted solution; L-infinity; bmo; EXISTENCE; uniqueness

摘要

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in R-p (p is an element of [1, infinity)) and backward stochastic differential equations (BSDEs) in R-p x H-p (p is an element of (1, infinity)) and in R-infinity x (L-infinity) over bar (BMO), with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman's inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multidimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Holder inequality for some suitable exponent p >= 1. Finally, some relations are established between Kazamaki's quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki's quadratic critical exponent of BMO martingales being infinite.