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A new Mertens decomposition of Yg,ξ-submartingale systems. Application to BSDEs with weak constraints at stopping times

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  • Dumitrescu, Roxana
  • Elie, Romuald
  • Sabbagh, Wissal
  • Zhou, Chao

Abstract

We first introduce the concept of Yg,ξ-submartingale systems, where the nonlinear operator Yg,ξ corresponds to the first component of the solution of a reflected BSDE with generator g and lower obstacle ξ. We first show that, in the case of a left-limited right-continuous obstacle, any Yg,ξ-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a Mertens decomposition, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. As an application, we introduce a new class of Backward Stochastic Differential Equations (in short BSDEs) with weak constraints at stopping times, which are related to the partial hedging of American options. We study the wellposedness of such equations and, using the Yg,ξ-Mertens decomposition, we show that the family of minimal time-t-values Yt, with (Y,Z) a supersolution of the BSDE with weak constraints, admits a representation in terms of a reflected backward stochastic differential equation.

Suggested Citation

  • Dumitrescu, Roxana & Elie, Romuald & Sabbagh, Wissal & Zhou, Chao, 2023. "A new Mertens decomposition of Yg,ξ-submartingale systems. Application to BSDEs with weak constraints at stopping times," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 183-205.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:183-205
    DOI: 10.1016/j.spa.2023.07.006
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    References listed on IDEAS

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    1. Klimsiak, Tomasz, 2015. "Reflected BSDEs on filtered probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4204-4241.
    2. Bruno Bouchard & Jean-François Chassagneux & Géraldine Bouveret, 2016. "A backward dual representation for the quantile hedging of Bermudan options," Post-Print hal-01069270, HAL.
    3. Philippe Briand & Romuald Elie & Ying Hu, 2018. "BSDEs with mean reflection," Post-Print hal-01318649, HAL.
    4. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2015. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Papers 1504.06094, arXiv.org, revised May 2017.
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