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2019 $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions
Shengjun Fan
Electron. J. Probab. 24(none): 1-48 (2019). DOI: 10.1214/19-EJP345

Abstract

We establish several existence, uniqueness and comparison results for $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumptions that the generator $g$ satisfies a one-sided Osgood condition together with a very general growth condition in $y$, a uniform continuity condition and/or a sub-linear growth condition in $z$, and a generalized Mokobodzki condition for reflected BSDEs which relates the growth of $g$ and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of $L^{1}$ solutions of the reflected BSDEs. We also prove that the $L^{1}$ solutions of reflected BSDEs can be approximated by a penalization method and by some sequences of $L^{1}$ solutions of reflected BSDEs. The approach is based on a combination between existing methods, their refinement and perfection, but also on some novel ideas and techniques. These results strengthen some existing work on the $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs.

Citation

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Shengjun Fan. "$L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions." Electron. J. Probab. 24 1 - 48, 2019. https://doi.org/10.1214/19-EJP345

Information

Received: 3 February 2018; Accepted: 19 July 2019; Published: 2019
First available in Project Euclid: 10 September 2019

zbMATH: 07107395
MathSciNet: MR4003141
Digital Object Identifier: 10.1214/19-EJP345

Subjects:
Primary: 60H10, 60H30

JOURNAL ARTICLE
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