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Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1860–1899 | Cite as

Existence, Uniqueness and Stability of \(L^1\) Solutions for Multidimensional Backward Stochastic Differential Equations with Generators of One-Sided Osgood Type

  • Sheng Jun Fan
Article
  • 124 Downloads

Abstract

We establish a general existence and uniqueness result of \(L^1\) solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator g satisfying a one-sided Osgood condition as well as a general growth condition in y, and a Lipschitz condition together with a sublinear growth condition in z, which improves some existing results. In particular, we put forward and prove a stability theorem of the \(L^1\) solutions for the first time. A new type of \(L^1\) solution is also investigated. Some delicate techniques involved in the relationship between convergence in \(L^1\) and in probability and dividing appropriately the time interval play crucial roles in our proofs.

Keywords

Backward stochastic differential equation \(L^1\) solution Existence and uniqueness Stability theorem One-sided Osgood condition 

Mathematics Subject Classification (2010)

60H10 

Notes

Acknowledgements

The author would like to thank the anonymous referee for his/her careful reading and helpful suggestions.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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