Harmonic analysis of stochastic equations and backward stochastic differential equations View Full Text


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Article Info

DATE

2008-12-12

AUTHORS

Freddy Delbaen, Shanjian Tang

ABSTRACT

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}^p}$$\end{document} (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p\in [1,\infty)}$$\end{document}) and backward stochastic differential equations (BSDEs) in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}^p\times \mathcal{H}^p}$$\end{document} (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p\in (1, \infty)}$$\end{document}) and in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$$\end{document} , 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 multi-dimensional 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 Hölder 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. More... »

PAGES

291

References to SciGraph publications

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  • 1976. On the existence and unicity of solutions of stochastic integral equations in PROBABILITY THEORY AND RELATED FIELDS
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