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Harmonic analysis of stochastic equations and backward stochastic differential equations
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  • Published: 12 December 2008

Harmonic analysis of stochastic equations and backward stochastic differential equations

  • Freddy Delbaen1 &
  • Shanjian Tang2 

Probability Theory and Related Fields volume 146, Article number: 291 (2010) Cite this article

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Abstract

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in \({\mathcal{R}^p}\) (\({p\in [1,\infty)}\)) and backward stochastic differential equations (BSDEs) in \({\mathcal{R}^p\times \mathcal{H}^p}\) (\({p\in (1, \infty)}\)) and in \({\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}\) , 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 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.

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Authors and Affiliations

  1. Department of Mathematics, Eidgenössische Technische Hochschule Zürich, 8092, Zürich, Switzerland

    Freddy Delbaen

  2. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, 200433, Shanghai, China

    Shanjian Tang

Authors
  1. Freddy Delbaen
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  2. Shanjian Tang
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Corresponding author

Correspondence to Shanjian Tang.

Additional information

Part of this work was done when F. Delbaen was visiting China in the years 2005, 2006 and 2007, Laboratory of Mathematics for Nonlinear Sciences, Fudan University, whose hospitality is greatly appreciated. Part of this work was financed by a grant of Credit-Suisse. The paper only reflects the personal opinion of the author. This work is partially supported by the NSFC under grant 10325101 (distinguished youth foundation), the Basic Research Program of China (973 Program) with grant no. 2007CB814904, and the Chang Jiang Scholars Program. Part of this work was completed when S. Tang was visiting in October, 2007, Department of Mathematics, Eidgenössische Technische Hochschule Zürich, whose hospitality is greatly appreciated.

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Delbaen, F., Tang, S. Harmonic analysis of stochastic equations and backward stochastic differential equations. Probab. Theory Relat. Fields 146, 291 (2010). https://doi.org/10.1007/s00440-008-0191-5

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  • Received: 08 March 2008

  • Revised: 09 October 2008

  • Published: 12 December 2008

  • DOI: https://doi.org/10.1007/s00440-008-0191-5

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Keywords

  • BMO martingales
  • Stochastic equations
  • Backward stochastic differential equations
  • Fefferman’s inequality
  • Reverse Hölder inequalities
  • Unbounded coefficients

Mathematics Subject Classification (2000)

  • Primary 60H10
  • 60H20
  • 60H99
  • Secondary 60G44
  • 60G46
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