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$$L^1$$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions. (English) Zbl 07107395
Summary: 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.
MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.)
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References:
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